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RD Sharma solutions for Class 11 Mathematics Textbook chapter 29 - Limits [Latest edition]

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Class 11 Mathematics Textbook - Shaalaa.com
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Chapter 29: Limits

Exercise 29.1Exercise 29.2Exercise 29.3Exercise 29.4Exercise 29.5Exercise 29.6Exercise 29.7Exercise 29.8Exercise 29.9Exercise 29.11Others
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Exercise 29.1 [Pages 11 - 23]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.1 [Pages 11 - 23]

Exercise 29.1 | Q 1 | Page 11

Show that \[\lim_{x \to 0} \frac{x}{\left| x \right|}\] does not exist.

Exercise 29.1 | Q 2 | Page 11

Find k so that \[\lim_{x \to 2} f\left( x \right)\] \[f\left( x \right) = \begin{cases}2x + 3, & x \leq 2 \\ x + k, & x > 2\end{cases} .\] 

Exercise 29.1 | Q 3 | Page 11

Show that \[\lim_{x \to 0} \frac{1}{x}\] does not exist. 

Exercise 29.1 | Q 4 | Page 11

Let f(x) be a function defined by \[f\left( x \right) = \begin{cases}\frac{3x}{\left| x \right| + 2x}, & x \neq 0 \\ 0, & x = 0\end{cases} .\] Show that \[\lim_{x \to 0} f\left( x \right)\] does not exist.

 
Exercise 29.1 | Q 5 | Page 11

Let \[f\left( x \right) = \left\{ \begin{array}{l}x + 1, & if x \geq 0 \\ x - 1, & if x < 0\end{array} . \right.\]Prove that \[\lim_{x \to 0} f\left( x \right)\] does not exist.

Exercise 29.1 | Q 6 | Page 11

Let \[f\left( x \right) = \begin{cases}x + 5, & if x > 0 \\ x - 4, & if x < 0\end{cases}\] \[\lim_{x \to 0} f\left( x \right)\]  does not exist. 

Exercise 29.1 | Q 7 | Page 11

Find \[\lim_{x \to 3} f\left( x \right)\] where \[f\left( x \right) = \begin{cases}4, & if x > 3 \\ x + 1, & if x < 3\end{cases}\] 

Exercise 29.1 | Q 8.1 | Page 11

If \[f\left( x \right) = \left\{ \begin{array}{l}2x + 3, & x \leq 0 \\ 3 \left( x + 1 \right), & x > 0\end{array} . \right.\] find \[\lim_{x \to 0} f\left( x \right)\] 

Exercise 29.1 | Q 8.2 | Page 11

If \[f\left(  x \right) = \left\{ \begin{array}{l}2x + 3, & x \leq 0 \\ 3 \left( x + 1 \right), & x > 0\end{array} . \right.\] find \[\lim_{x \to 1} f\left( x \right)\]

Exercise 29.1 | Q 9 | Page 11

Find \[\lim_{x \to 1} f\left( x \right)\] if \[f\left( x \right) = \begin{cases}x^2 - 1, & x \leq 1 \\ - x^2 - 1, & x > 1\end{cases}\] 

Exercise 29.1 | Q 10 | Page 11

Evaluate \[\lim_{x \to 0} f\left( x \right)\]  where \[f\left( x \right) = \begin{cases}\frac{\left| x \right|}{x}, & x \neq 0 \\ 0, & x = 0\end{cases}\] 

Exercise 29.1 | Q 11 | Page 11

Let a1a2, ..., an be fixed real numbers such that
f(x) = (x − a1) (x − a2) ... (x − an)
What is \[\lim_{x \to a_1} f\left( x \right)?\] Compute \[\lim_{x \to a} f\left( x \right) .\] 

Exercise 29.1 | Q 12 | Page 11

Find \[\lim_{x \to 1^+} \left( \frac{1}{x - 1} \right) .\] 

Exercise 29.1 | Q 13.01 | Page 11

Evaluate the following one sided limit: 

\[\lim_{x \to 2^+} \frac{x - 3}{x^2 - 4}\] 

Exercise 29.1 | Q 13.02 | Page 11

Evaluate the following one sided limit: 

\[\lim_{x \to 2^-} \frac{x - 3}{x^2 - 4}\] 

Exercise 29.1 | Q 13.03 | Page 11

Evaluate the following one sided limit:

\[\lim_{x \to 0^+} \frac{1}{3x}\]

Exercise 29.1 | Q 13.04 | Page 11

Evaluate the following one sided limit:

\[\lim_{x \to - 8^+} \frac{2x}{x + 8}\]

Exercise 29.1 | Q 13.05 | Page 11

Evaluate the following one sided limit:

\[\lim_{x \to 0^+} \frac{2}{x^{1/5}}\]

Exercise 29.1 | Q 13.06 | Page 11

Evaluate the following one sided limit: 

\[\lim_{x \to \frac{\pi}{2}} \tan x\]

Exercise 29.1 | Q 13.07 | Page 11

Evaluate the following one sided limit:

\[\lim_{x \to \frac{\pi}{2}} \tan x\]

Exercise 29.1 | Q 13.08 | Page 11

Evaluate the following one sided limit:

\[\lim_{x \to 0^-} \frac{x^2 - 3x + 2}{x^3 - 2 x^2}\]

Exercise 29.1 | Q 13.09 | Page 11

Evaluate the following one sided limit:

\[\lim_{x \to - 2^+} \frac{x^2 - 1}{2x + 4}\]

Exercise 29.1 | Q 13.1 | Page 11

Evaluate the following one sided limit:

\[\lim_{x \to 0^-} 2 - \cot x\] 

Exercise 29.1 | Q 13.11 | Page 11

Evaluate the following one sided limit:

\[\lim_{x \to 0^-} 1 + cosec x\]

Exercise 29.1 | Q 14 | Page 12

Show that \[\lim_{x \to 0} e^{- 1/x}\] does not exist. 

Exercise 29.1 | Q 15.1 | Page 12

Find: \[\ \lim_{x \to 2} \left[ x \right]\] 

Exercise 29.1 | Q 15.2 | Page 12

Find: \[ \lim_{x \to \frac{5}{2}} \left[ x \right]\]

 

Exercise 29.1 | Q 15.3 | Page 12

Find: \[ \lim_{x \to 1} \left[ x \right]\]

Exercise 29.1 | Q 16 | Page 12

Prove that \[\lim_{x \to a^+} \left[ x \right] = \left[ a \right]\] R. Also, prove that \[\lim_{x \to 1^-} \left[ x \right] = 0 .\]

Exercise 29.1 | Q 17 | Page 12

Show that \[\lim_{x \to 2^-} \frac{x}{\left[ x \right]} \neq \lim_{x \to 2^+} \frac{x}{\left[ x \right]} .\]

Exercise 29.1 | Q 18 | Page 12

Find \[\lim_{x \to 3^+} \frac{x}{\left[ x \right]} .\]  Is it equal to \[\lim_{x \to 3^-} \frac{x}{\left[ x \right]} .\]

Exercise 29.1 | Q 19 | Page 12

Find \[\lim_{x \to 5/2} \left[ x \right] .\] 

Exercise 29.1 | Q 20 | Page 12

Evaluate \[\lim_{x \to 2} f\left( x \right)\] (if it exists), where \[f\left( x \right) = \left\{ \begin{array}{l}x - \left[ x \right], & x < 2 \\ 4, & x = 2 \\ 3x - 5, & x > 2\end{array} . \right.\]

Exercise 29.1 | Q 21 | Page 12

Show that \[\lim_{x \to 0} \sin \frac{1}{x}\]does not exist. 

Exercise 29.1 | Q 21 | Page 23

\[\lim_{x \to 2} \left( \frac{1}{x - 2} - \frac{4}{x^3 - 2 x^2} \right)\] 

Exercise 29.1 | Q 22 | Page 12

Let \[f\left( x \right) = \begin{cases}\frac{k\cos x}{\pi - 2x}, & where x \neq \frac{\pi}{2} \\ 3, & where x = \frac{\pi}{2}\end{cases}\]   and if \[\lim_{x \to \frac{\pi}{2}} f\left( x \right) = f\left( \frac{\pi}{2} \right)\] 

Exercise 29.2 [Page 18]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.2 [Page 18]

Exercise 29.2 | Q 1 | Page 18

\[\lim_{x \to 1} \frac{x^2 + 1}{x + 1}\] 

Exercise 29.2 | Q 2 | Page 18

\[\lim_{x \to 0} \frac{2 x^2 + 3x + 4}{x^2 + 3x + 2}\] 

Exercise 29.2 | Q 3 | Page 18

\[\lim_{x \to 3} \frac{\sqrt{2x + 3}}{x + 3}\] 

Exercise 29.2 | Q 4 | Page 18

\[\lim_{x \to 1} \frac{\sqrt{x + 8}}{\sqrt{x}}\] 

Exercise 29.2 | Q 5 | Page 18

\[\lim_{x \to a} \frac{\sqrt{x} + \sqrt{a}}{x + a}\] 

Exercise 29.2 | Q 6 | Page 18

\[\lim_{x \to 1} \frac{1 + \left( x - 1 \right)^2}{1 + x^2}\]

Exercise 29.2 | Q 7 | Page 18

\[\lim_{x \to 0} \frac{x^{2/3} - 9}{x - 27}\]

Exercise 29.2 | Q 8 | Page 18

\[\lim_{x \to 0} 9\] 

Exercise 29.2 | Q 9 | Page 18

\[\lim_{x \to 2} \left( 3 - x \right)\] 

Exercise 29.2 | Q 10 | Page 18

\[\lim_{x \to - 1}{\left( 4 x^2 + 2 \right)}\]

Exercise 29.2 | Q 11 | Page 18

\[\lim_{x \to - 1} \frac{x^3 - 3x + 1}{x - 1}\]

Exercise 29.2 | Q 12 | Page 18

\[\lim_{x \to 0} \frac{3x + 1}{x + 3}\] 

Exercise 29.2 | Q 13 | Page 18

\[\lim_{x \to 3} \frac{x^2 - 9}{x + 2}\] 

Exercise 29.2 | Q 14 | Page 18

\[\lim_{x \to 0} \frac{ax + b}{cx + d}, d \neq 0\]

Exercise 29.3 [Pages 23 - 24]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.3 [Pages 23 - 24]

Exercise 29.3 | Q 1 | Page 23

\[\lim_{x \to - 5} \frac{2 x^2 + 9x - 5}{x + 5}\] 

Exercise 29.3 | Q 2 | Page 23

\[\lim_{x \to 3} \frac{x^2 - 4x + 3}{x^2 - 2x - 3}\] 

Exercise 29.3 | Q 3 | Page 23

\[\lim_{x \to 3} \frac{x^4 - 81}{x^2 - 9}\] 

Exercise 29.3 | Q 4 | Page 23

\[\lim_{x \to 2} \frac{x^3 - 8}{x^2 - 4}\] 

Exercise 29.3 | Q 5 | Page 23

\[\lim_{x \to - 1/2} \frac{8 x^3 + 1}{2x + 1}\] 

Exercise 29.3 | Q 6 | Page 23

\[\lim_{x \to 4} \frac{x^2 - 7x + 12}{x^2 - 3x - 4}\] 

Exercise 29.3 | Q 7 | Page 23

\[\lim_{x \to 2} \frac{x^4 - 16}{x - 2}\] 

Exercise 29.3 | Q 8 | Page 23

\[\lim_{x \to 5} \frac{x^2 - 9x + 20}{x^2 - 6x + 5}\] 

Exercise 29.3 | Q 9 | Page 23

\[\lim_{x \to - 1} \frac{x^3 + 1}{x + 1}\] 

Exercise 29.3 | Q 10 | Page 23

\[\lim_{x \to 5} \frac{x^3 - 125}{x^2 - 7x + 10}\] 

Exercise 29.3 | Q 11 | Page 23

\[\lim_{x \to \sqrt{2}} \frac{x^2 - 2}{x^2 + \sqrt{2}x - 4}\]

Exercise 29.3 | Q 12 | Page 23

\[\lim_{x \to \sqrt{3}} \frac{x^2 - 3}{x^2 + 3 \sqrt{3}x - 12}\]

Exercise 29.3 | Q 13 | Page 23

\[\lim_{x \to \sqrt{3}} \frac{x^4 - 9}{x^2 + 4\sqrt{3}x - 15}\]

Exercise 29.3 | Q 14 | Page 23

\[\lim_{x \to 2} \left( \frac{x}{x - 2} - \frac{4}{x^2 - 2x} \right)\] 

Exercise 29.3 | Q 15 | Page 23

\[\lim_{x \to 1} \left( \frac{1}{x^2 + x - 2} - \frac{x}{x^3 - 1} \right)\] 

Exercise 29.3 | Q 16 | Page 23

\[\lim_{x \to \sqrt{3}} \frac{x^4 - 9}{x^2 + 4\sqrt{3}x - 15}\]

Exercise 29.3 | Q 17 | Page 23

\[\lim_{x \to 2} \left( \frac{1}{x - 2} - \frac{2}{x^2 - 2x} \right)\] 

Exercise 29.3 | Q 18 | Page 23

\[\lim_{x \to 1/4} \frac{4x - 1}{2\sqrt{x} - 1}\] 

Exercise 29.3 | Q 19 | Page 23

\[\lim_{x \to 4} \frac{x^2 - 16}{\sqrt{x} - 2}\] 

Exercise 29.3 | Q 20 | Page 23

\[\lim_{x \to 0} \frac{\left( a + x \right)^2 - a^2}{x}\] 

Exercise 29.3 | Q 21 | Page 23
\[\lim_{x \to 3} \left( \frac{1}{x - 3} - \frac{3}{x^2 - 3x} \right)\]
Exercise 29.3 | Q 22 | Page 23

\[\lim_{x \to 3} \left( \frac{1}{x - 3} - \frac{3}{x^2 - 3x} \right)\] 

Exercise 29.3 | Q 23 | Page 23

\[\lim_{x \to 1} \left( \frac{1}{x - 1} - \frac{2}{x^2 - 1} \right)\]

Exercise 29.3 | Q 24 | Page 23

\[\lim_{x \to 3} \left( x^2 - 9 \right) \left[ \frac{1}{x + 3} + \frac{1}{x - 3} \right]\] 

Exercise 29.3 | Q 25 | Page 23

\[\lim_{x \to 1} \frac{x^4 - 3 x^3 + 2}{x^3 - 5 x^2 + 3x + 1}\] 

Exercise 29.3 | Q 26 | Page 23

\[\lim_{x \to 2} \frac{x^3 + 3 x^2 - 9x - 2}{x^3 - x - 6}\] 

Exercise 29.3 | Q 27 | Page 23

\[\lim_{x \to 1} \frac{1 - x^{- 1/3}}{1 - x^{- 2/3}}\] 

Exercise 29.3 | Q 28 | Page 23

\[\lim_{x \to 3} \frac{x^2 - x - 6}{x^3 - 3 x^2 + x - 3}\]

Exercise 29.3 | Q 29 | Page 23

\[\lim_{x \to - 2} \frac{x^3 + x^2 + 4x + 12}{x^3 - 3x + 2}\]

Exercise 29.3 | Q 30 | Page 23

\[\lim_{x \to 1} \frac{x^3 + 3 x^2 - 6x + 2}{x^3 + 3 x^2 - 3x - 1}\]

Exercise 29.3 | Q 31 | Page 24

\[\lim_{x \to 2} \left[ \frac{1}{x - 2} - \frac{2\left( 2x - 3 \right)}{x^3 - 3 x^2 + 2x} \right]\] 

Exercise 29.3 | Q 32 | Page 24

\[\lim_{x \to 1} \frac{\sqrt{x^2 - 1} + \sqrt{x - 1}}{\sqrt{x^2 - 1}}, x > 1\] 

Exercise 29.3 | Q 33 | Page 24

\[\lim_{x \to 1} \left\{ \frac{x - 2}{x^2 - x} - \frac{1}{x^3 - 3 x^2 + 2x} \right\}\] 

Exercise 29.3 | Q 34 | Page 24

Evaluate the following limit:

\[\lim_{x \to 1} \frac{x^7 - 2 x^5 + 1}{x^3 - 3 x^2 + 2}\] 

Exercise 29.4 [Pages 28 - 30]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.4 [Pages 28 - 30]

Exercise 29.4 | Q 1 | Page 28

\[\lim_{x \to 0} \frac{\sqrt{1 + x + x^2} - 1}{x}\]

Exercise 29.4 | Q 2 | Page 28

\[\lim_{x \to 0} \frac{2x}{\sqrt{a + x} - \sqrt{a - x}}\] 

Exercise 29.4 | Q 3 | Page 28

\[\lim_{x \to 0} \frac{\sqrt{a^2 + x^2} - a}{x^2}\] 

Exercise 29.4 | Q 4 | Page 28

\[\lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{2x}\]

Exercise 29.4 | Q 5 | Page 28

\[\lim_{x \to 2} \frac{\sqrt{3 - x} - 1}{2 - x}\] 

Exercise 29.4 | Q 6 | Page 28

\[\lim_{x \to 3} \frac{x - 3}{\sqrt{x - 2} - \sqrt{4 - x}}\] 

Exercise 29.4 | Q 7 | Page 28

\[\lim_{x \to 0} \frac{x}{\sqrt{1 + x} - \sqrt{1 - x}}\] 

Exercise 29.4 | Q 8 | Page 28

\[\lim_{x \to 1} \frac{\sqrt{5x - 4} - \sqrt{x}}{x - 1}\] 

Exercise 29.4 | Q 9 | Page 28

\[\lim_{x \to 1} \frac{x - 1}{\sqrt{x^2 + 3 - 2}}\] 

Exercise 29.4 | Q 10 | Page 28

\[\lim_{x \to 3} \frac{\sqrt{x + 3} - \sqrt{6}}{x^2 - 9}\] 

Exercise 29.4 | Q 11 | Page 28

\[\lim_{x \to 1} \frac{\sqrt{5x - 4} - \sqrt{x}}{x^2 - 1}\] 

Exercise 29.4 | Q 12 | Page 28

\[\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}\] 

Exercise 29.4 | Q 13 | Page 28

\[\lim_{x \to 2} \frac{\sqrt{x^2 + 1} - \sqrt{5}}{x - 2}\] 

Exercise 29.4 | Q 14 | Page 28

\[\lim_{x \to 2} \frac{x - 2}{\sqrt{x} - \sqrt{2}}\] 

Exercise 29.4 | Q 15 | Page 28

\[\lim_{x \to 7} \frac{4 - \sqrt{9 + x}}{1 - \sqrt{8 - x}}\] 

Exercise 29.4 | Q 16 | Page 28

\[\lim_{x \to 0} \frac{\sqrt{a + x} - \sqrt{a}}{x\sqrt{a^2 + ax}}\]

Exercise 29.4 | Q 17 | Page 28

\[\lim_{x \to 5} \frac{x - 5}{\sqrt{6x - 5} - \sqrt{4x + 5}}\] 

Exercise 29.4 | Q 18 | Page 28

\[\lim_{x \to 1} \frac{\sqrt{5x - 4} - \sqrt{x}}{x^3 - 1}\] 

Exercise 29.4 | Q 19 | Page 28

\[\lim_{x \to 2} \frac{\sqrt{1 + 4x} - \sqrt{5 + 2x}}{x - 2}\] 

Exercise 29.4 | Q 20 | Page 28

\[\lim_{x \to 1} \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^2 - 1}\] 

Exercise 29.4 | Q 21 | Page 29

\[\lim_{x \to 0} \frac{\sqrt{1 + x^2} - \sqrt{1 - x^2}}{x}\] 

Exercise 29.4 | Q 22 | Page 29

\[\lim_{x \to 0} \frac{\sqrt{1 + x + x^2} - \sqrt{x + 1}}{2 x^2}\] 

Exercise 29.4 | Q 23 | Page 29

\[\lim_{x \to 4} \frac{2 - \sqrt{x}}{4 - x}\]

Exercise 29.4 | Q 24 | Page 29

\[\lim_{x \to a} \frac{x - a}{\sqrt{x} - \sqrt{a}}\]

Exercise 29.4 | Q 25 | Page 29

\[\lim_{x \to 0} \frac{\sqrt{1 + 3x} - \sqrt{1 - 3x}}{x}\]

Exercise 29.4 | Q 26 | Page 29

\[\lim_{x \to 0} \frac{\sqrt{2 - x} - \sqrt{2 + x}}{x}\] 

Exercise 29.4 | Q 27 | Page 29

\[\lim_{x \to 1} \frac{\sqrt{3 + x} - \sqrt{5 - x}}{x^2 - 1}\] 

Exercise 29.4 | Q 28 | Page 29

\[\lim_{x \to 1} \frac{\left( 2x - 3 \right) \left( \sqrt{x} - 1 \right)}{3 x^2 + 3x - 6}\]

Exercise 29.4 | Q 29 | Page 29

\[\lim_{x \to 0} \frac{\sqrt{1 + x^2} - \sqrt{1 + x}}{\sqrt{1 + x^3} - \sqrt{1 + x}}\] 

Exercise 29.4 | Q 29 | Page 30

\[\lim_{x \to 1} \frac{ x^2 - \sqrt{x}}{\sqrt{x} - 1}\]

Exercise 29.4 | Q 31 | Page 29

\[\lim_{h \to 0} \frac{\sqrt{x + h} - \sqrt{x}}{h}, x \neq 0\] 

Exercise 29.4 | Q 32 | Page 29

\[\lim_{x \to \sqrt{10}} \frac{\sqrt{7 + 2x} - \left( \sqrt{5} + \sqrt{2} \right)}{x^2 - 10}\] 

Exercise 29.4 | Q 33 | Page 29

\[\lim_{x \to \sqrt{6}} \frac{\sqrt{5 + 2x} - \left( \sqrt{3} + \sqrt{2} \right)}{x^2 - 6}\] 

 

Exercise 29.4 | Q 34 | Page 29

\[\lim_{x \to \sqrt{2}} \frac{\sqrt{3 + 2x} - \left( \sqrt{2} + 1 \right)}{x^2 - 2}\] 

Exercise 29.5 [Page 33]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.5 [Page 33]

Exercise 29.5 | Q 1 | Page 33

\[\lim_{x \to a} \frac{\left( x + 2 \right)^{5/2} - \left( a + 2 \right)^{5/2}}{x - a}\] 

Exercise 29.5 | Q 2 | Page 33

\[\lim_{x \to a} \frac{\left( x + 2 \right)^{3/2} - \left( a + 2 \right)^{3/2}}{x -  a}\]

Exercise 29.5 | Q 3 | Page 33

\[\lim_{x \to 0} \frac{\left( 1 + x \right)^6 - 1}{\left( 1 + x \right)^2 - 1}\] 

Exercise 29.5 | Q 4 | Page 33

\[\lim_{x \to a} \frac{x^{2/7} - a^{2/7}}{x - a}\] 

Exercise 29.5 | Q 5 | Page 33

\[\lim_{x \to a} \frac{x^{5/7} - a^{5/7}}{x^{2/7} - a^{2/7}}\] 

Exercise 29.5 | Q 6 | Page 33

\[\lim_{x \to - 1/2} \frac{8 x^3 + 1}{2x + 1}\]

Exercise 29.5 | Q 7 | Page 33

\[\lim_{x \to 27} \frac{\left( x^{1/3} + 3 \right) \left( x^{1/3} - 3 \right)}{x - 27}\] 

Exercise 29.5 | Q 8 | Page 33

\[\lim_{x \to 4} \frac{x^3 - 64}{x^2 - 16}\] 

Exercise 29.5 | Q 9 | Page 33

\[\lim_{x \to 1} \frac{x^{15} - 1}{x^{10} - 1}\] 

Exercise 29.5 | Q 10 | Page 33

\[\lim_{x \to - 1} \frac{x^3 + 1}{x + 1}\] 

Exercise 29.5 | Q 11 | Page 33

\[\lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x^{3/4} - a^{3/4}}\] 

Exercise 29.5 | Q 12 | Page 33

If \[\lim_{x \to 3} \frac{x^n - 3^n}{x - 3} = 108,\]  find the value of n

Exercise 29.5 | Q 13 | Page 33

If \[\lim_{x \to a} \frac{x^9 - a^9}{x - a} = 9,\] find all possible values of a

Exercise 29.5 | Q 14 | Page 33

If \[\lim_{x \to a} \frac{x^5 - a^5}{x - a} = 405,\]find all possible values of a

 

 

Exercise 29.5 | Q 15 | Page 33

If \[\lim_{x \to a} \frac{x^9 - a^9}{x - a} = \lim_{x \to 5} \left( 4 + x \right),\] find all possible values of a

Exercise 29.5 | Q 16 | Page 33

If \[\lim_{x \to a} \frac{x^3 - a^3}{x - a} = \lim_{x \to 1} \frac{x^4 - 1}{x - 1},\] find all possible values of a

Exercise 29.6 [Pages 38 - 39]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.6 [Pages 38 - 39]

Exercise 29.6 | Q 1 | Page 38

\[\lim_{x \to \infty} \frac{\left( 3x - 1 \right) \left( 4x - 2 \right)}{\left( x + 8 \right) \left( x - 1 \right)}\] 

Exercise 29.6 | Q 2 | Page 38

\[\lim_{x \to \infty} \frac{3 x^3 - 4 x^2 + 6x - 1}{2 x^3 + x^2 - 5x + 7}\] 

Exercise 29.6 | Q 3 | Page 38

\[\lim_{x \to \infty} \frac{5 x^3 - 6}{\sqrt{9 + 4 x^6}}\]

Exercise 29.6 | Q 4 | Page 38

\[\lim_{x \to \infty} \sqrt{x^2 + cx - x}\] 

Exercise 29.6 | Q 5 | Page 38

\[\lim_{x \to \infty} \sqrt{x + 1} - \sqrt{x}\] 

Exercise 29.6 | Q 6 | Page 38

\[\lim_{x \to \infty} \sqrt{x^2 + 7x - x}\] 

Exercise 29.6 | Q 7 | Page 38

\[\lim_{x \to \infty} \frac{x}{\sqrt{4 x^2 + 1} - 1}\] 

Exercise 29.6 | Q 8 | Page 38

\[\lim_{n \to \infty} \frac{n^2}{1 + 2 + 3 + . . . + n}\] 

Exercise 29.6 | Q 9 | Page 39

\[\lim_{x \to \infty} \frac{3 x^{- 1} + 4 x^{- 2}}{5 x^{- 1} + 6 x^{- 2}}\]

Exercise 29.6 | Q 10 | Page 39

\[\lim_{x \to \infty} \frac{\sqrt{x^2 + a^2} - \sqrt{x^2 + b^2}}{\sqrt{x^2 + c^2} - \sqrt{x^2 + d^2}}\] 

Exercise 29.6 | Q 11 | Page 39

\[\lim_{n \to \infty} \left[ \frac{\left( n + 2 \right)! + \left( n + 1 \right)!}{\left( n + 2 \right)! - \left( n + 1 \right)!} \right]\] 

Exercise 29.6 | Q 12 | Page 39

\[\lim_{x \to \infty} \left[ x\left\{ \sqrt{x^2 + 1} - \sqrt{x^2 - 1} \right\} \right]\] 

Exercise 29.6 | Q 13 | Page 39

\[\lim_{x \to \infty} \left[ \left\{ \sqrt{x + 1} - \sqrt{x} \right\} \sqrt{x + 2} \right]\] 

Exercise 29.6 | Q 14 | Page 39

\[\lim_{n \to \infty} \left[ \frac{1^2 + 2^2 + . . . + n^2}{n^3} \right]\]

Exercise 29.6 | Q 15 | Page 39

\[\lim_{n \to \infty} \left[ \frac{1 + 2 + 3 . . . . . . n - 1}{n^2} \right]\] 

Exercise 29.6 | Q 16 | Page 39

\[\lim_{n \to \infty} \left[ \frac{1^3 + 2^3 + . . . . n^3}{n^4} \right]\]

Exercise 29.6 | Q 17 | Page 39

\[\lim_{n \to \infty} \left[ \frac{1^3 + 2^3 + . . . n^3}{\left( n - 1 \right)^4} \right]\] 

Exercise 29.6 | Q 18 | Page 39

\[\lim_{x \to \infty} \left[ \sqrt{x}\left\{ \sqrt{x + 1} - \sqrt{x} \right\} \right]\] 

Exercise 29.6 | Q 19 | Page 39

\[\lim_{n \to \infty} \left[ \frac{1}{3} + \frac{1}{3^2} + \frac{1}{3^3} + . . . + \frac{1}{3^n} \right]\] 

Exercise 29.6 | Q 20 | Page 39

\[\lim_{x \to \infty} \left[ \frac{x^4 + 7 x^3 + 46x + a}{x^4 + 6} \right]\] where a is a non-zero real number. 

Exercise 29.6 | Q 21 | Page 39

\[f\left( x \right) = \frac{a x^2 + b}{x^2 + 1}, \lim_{x \to 0} f\left( x \right) = 1\] and \[\lim_{x \to \infty} f\left( x \right) = 1,\]then prove that f(−2) = f(2) = 1

Exercise 29.6 | Q 22 | Page 39

Show that \[\lim_{x \to \infty} \left( \sqrt{x^2 + x + 1} - x \right) \neq \lim_{x \to \infty} \left( \sqrt{x^2 + 1} - x \right)\] 

Exercise 29.6 | Q 23 | Page 39

\[\lim_{x \to - \infty} \left( \sqrt{4 x^2 - 7x} + 2x \right)\] 

Exercise 29.6 | Q 24 | Page 39

\[\lim_{x \to - \infty} \left( \sqrt{x^2 - 8x} + x \right)\] 

Exercise 29.6 | Q 25 | Page 39

Evaluate: \[\lim_{n \to \infty} \frac{1^4 + 2^4 + 3^4 + . . . + n^4}{n^5} - \lim_{n \to \infty} \frac{1^3 + 2^3 + . . . + n^3}{n^5}\] 

Exercise 29.6 | Q 26 | Page 39

Evaluate: \[\lim_{n \to \infty} \frac{1 . 2 + 2 . 3 + 3 . 4 + . . . + n\left( n + 1 \right)}{n^3}\] 

Exercise 29.7 [Pages 49 - 51]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.7 [Pages 49 - 51]

Exercise 29.7 | Q 1 | Page 49

\[\lim_{x \to 0} \frac{\sin 3x}{5x}\] 

Exercise 29.7 | Q 2 | Page 49

\[\lim_{x \to 0} \frac{\sin x^0}{x}\] 

Exercise 29.7 | Q 3 | Page 49

\[\lim_{x \to 0} \left[ \frac{x^2}{\sin x^2} \right]\] 

Exercise 29.7 | Q 4 | Page 49

\[\lim_{x \to 0} \frac{\sin x \cos x}{3x}\] 

Exercise 29.7 | Q 5 | Page 50

\[\lim_{x \to 0} \frac{3 \sin x - 4 \sin^3 x}{x}\] 

Exercise 29.7 | Q 6 | Page 50

\[\lim_{x \to 0} \frac{\tan 8x}{\sin 2x}\] 

Exercise 29.7 | Q 7 | Page 50

\[\lim_{x \to 0} \frac{\tan mx}{\tan nx}\] 

Exercise 29.7 | Q 8 | Page 50

\[\lim_{x \to 0} \frac{\sin 5x}{\tan 3x}\] 

Exercise 29.7 | Q 9 | Page 50

\[\lim_{x \to 0} \frac{\sin x^n}{x^n}\] 

Exercise 29.7 | Q 10 | Page 50

\[\lim_{x \to 0} \frac{7x \cos x - 3 \sin x}{4x + \tan x}\] 

Exercise 29.7 | Q 11 | Page 50

\[\lim_{x \to 0} \frac{\cos ax - \cos bx}{\cos cx - \cos dx}\] 

Exercise 29.7 | Q 12 | Page 50

\[\lim_{x \to 0} \frac{\tan^2 3x}{x^2}\] 

Exercise 29.7 | Q 13 | Page 50

\[\lim_{x \to 0} \frac{1 - \cos mx}{x^2}\] 

Exercise 29.7 | Q 14 | Page 50

\[\lim_{x \to 0} \frac{3 \sin 2x + 2x}{3x + 2 \tan 3x}\] 

Exercise 29.7 | Q 15 | Page 50

\[\lim_{x \to 0} \frac{\cos 3x - \cos 7x}{x^2}\] 

Exercise 29.7 | Q 16 | Page 50

\[\lim_\theta \to 0 \frac{\sin 3\theta}{\tan 2\theta}\] 

Exercise 29.7 | Q 17 | Page 50

\[\lim_{x \to 0} \frac{\sin x^2 \left( 1 - \cos x^2 \right)}{x^6}\] 

Exercise 29.7 | Q 18 | Page 50

\[\lim_{x \to 0} \frac{\sin^2 4 x^2}{x^4}\] 

Exercise 29.7 | Q 19 | Page 50

\[\lim_{x \to 0} \frac{x \cos x + 2 \sin x}{x^2 + \tan x}\] 

Exercise 29.7 | Q 20 | Page 50

\[\lim_{x \to 0} \frac{2x - \sin x}{\tan x + x}\] 

Exercise 29.7 | Q 21 | Page 50

\[\lim_{x \to 0} \frac{5 x \cos x + 3 \sin x}{3 x^2 + \tan x}\] 

Exercise 29.7 | Q 22 | Page 50

\[\lim_{x \to 0} \frac{\sin 3x - \sin x}{\sin x}\] 

Exercise 29.7 | Q 23 | Page 50

\[\lim_{x \to 0} \frac{\sin 5x - \sin 3x}{\sin x}\] 

Exercise 29.7 | Q 24 | Page 50
\[\lim_{x \to 0} \frac{\cos 3x - \cos 5x}{x^2}\]
Exercise 29.7 | Q 25 | Page 50

\[\lim_{x \to 0} \frac{\tan 3x - 2x}{3x - \sin^2 x}\] 

Exercise 29.7 | Q 26 | Page 50

\[\lim_{x \to 0} \frac{\sin \left( 2 + x \right) - \sin \left( 2 - x \right)}{x}\]

Exercise 29.7 | Q 27 | Page 50

\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin \left( a + h \right) - a^2 \sin a}{h}\] 

Exercise 29.7 | Q 28 | Page 50

\[\lim_{x \to 0} \frac{\tan x - \sin x}{\sin 3x - 3 \sin x}\]

Exercise 29.7 | Q 29 | Page 50

\[\lim_{x \to 0} \frac{\sec 5x - \sec 3x}{\sec 3x - \sec x}\]

Exercise 29.7 | Q 30 | Page 50

\[\lim_{x \to 0} \frac{1 - \cos 2x}{\cos 2x - \cos 8x}\]

Exercise 29.7 | Q 31 | Page 50

\[\lim_{x \to 0} \frac{1 - \cos 2x + \tan^2 x}{x \sin x}\] 

Exercise 29.7 | Q 32 | Page 50

\[\lim_{x \to 0} \frac{\sin \left( a + x \right) + \sin \left( a - x \right) - 2 \sin a}{x \sin x}\] 

Exercise 29.7 | Q 33 | Page 50

\[\lim_{x \to 0} \frac{x^2 - \tan 2x}{\tan x}\] 

Exercise 29.7 | Q 34 | Page 50

\[\lim_{x \to 0} \frac{\sqrt{2} - \sqrt{1 + \cos x}}{x^2}\] 

Exercise 29.7 | Q 35 | Page 50

\[\lim_{x \to 0} \frac{x \tan x}{1 - \cos x}\]

Exercise 29.7 | Q 36 | Page 50

\[\lim_{x \to 0} \frac{x^2 + 1 - \cos x}{x \sin x}\] 

Exercise 29.7 | Q 37 | Page 50

\[\lim_{x \to 0} \frac{\sin 2x \left( \cos 3x - \cos x \right)}{x^3}\] 

Exercise 29.7 | Q 38 | Page 50

\[\lim_{x \to 0} \frac{2 \sin x^\circ - \sin 2 x^\circ}{x^3}\] 

Exercise 29.7 | Q 39 | Page 50

\[\lim_{x \to 0} \frac{x^3 \cot x}{1 - \cos x}\] 

Exercise 29.7 | Q 40 | Page 50

\[\lim_{x \to 0} \frac{x \tan x}{1 - \cos 2x}\] 

Exercise 29.7 | Q 41 | Page 50

\[\lim_{x \to 0} \frac{\sin \left( 3 + x \right) - \sin \left( 3 - x \right)}{x}\] 

Exercise 29.7 | Q 42 | Page 50

\[\lim_{x \to 0} \frac{\cos 2x - 1}{\cos x - 1}\] 

Exercise 29.7 | Q 43 | Page 51

\[\lim_{x \to 0} \frac{3 \sin^2 x - 2 \sin x^2}{3 x^2}\] 

Exercise 29.7 | Q 44 | Page 51

\[\lim_{x \to 0} \frac{\sqrt{1 + \sin x} - \sqrt{1 - \sin x}}{x}\] 

Exercise 29.7 | Q 45 | Page 51

\[\lim_{x \to 0} \frac{1 - \cos 4x}{x^2}\] 

Exercise 29.7 | Q 46 | Page 51

\[\lim_{x \to 0} \frac{x \cos x + \sin x}{x^2 + \tan x}\] 

Exercise 29.7 | Q 47 | Page 51

\[\lim_{x \to 0} \frac{1 - \cos 2x}{3 \tan^2 x}\] 

Exercise 29.7 | Q 48 | Page 51

\[\lim_\theta \to 0 \frac{1 - \cos 4\theta}{1 - \cos 6\theta}\] 

Exercise 29.7 | Q 49 | Page 51

\[\lim_{x \to 0} \frac{ax + x \cos x}{b \sin x}\]

Exercise 29.7 | Q 50 | Page 51

\[\lim_\theta \to 0 \frac{\sin 4\theta}{\tan 3\theta}\] 

Exercise 29.7 | Q 51 | Page 51

Evaluate the following limits: 

\[\lim_{x \to 0} \frac{2\sin x - \sin2x}{x^3}\] 

 

Exercise 29.7 | Q 52 | Page 51

\[\lim_{x \to 0} \frac{1 - \cos 5x}{1 - \cos 6x}\]

Exercise 29.7 | Q 53 | Page 51

\[\lim_{x \to 0} \frac{cosec x - \cot x}{x}\]

Exercise 29.7 | Q 54 | Page 51

\[\lim_{x \to 0} \frac{\sin 3x + 7x}{4x + \sin 2x}\]

Exercise 29.7 | Q 55 | Page 51

\[\lim_{x \to 0} \frac{5x + 4 \sin 3x}{4 \sin 2x + 7x}\]

Exercise 29.7 | Q 56 | Page 51

\[\lim_{x \to 0} \frac{3 \sin x - \sin 3x}{x^3}\]

Exercise 29.7 | Q 57 | Page 51

\[\lim_{x \to 0} \frac{\tan 2x - \sin 2x}{x^3}\]

Exercise 29.7 | Q 58 | Page 51

\[\lim_{x \to 0} \frac{\sin ax + bx}{ax + \sin bx}\]

Exercise 29.7 | Q 59 | Page 51

\[\lim_{x \to 0} \left( cosec x - \cot x \right)\]

Exercise 29.7 | Q 60 | Page 51

Evaluate the following limit: 

\[\lim_{x \to 0} \frac{\sin\left( \alpha + \beta \right)x + \sin\left( \alpha - \beta \right)x + \sin2\alpha x}{\cos^2 \beta x - \cos^2 \alpha x}\]

Exercise 29.7 | Q 61 | Page 51

Evaluate the following limits: 

\[\lim_{x \to 0} \frac{\cos ax - \cos bx}{\cos cx - 1}\] 

Exercise 29.7 | Q 62 | Page 51

Evaluate the following limit: 

\[\lim_{h \to 0} \frac{\left( a + h \right)^2 \sin\left( a + h \right) - a^2 \sin a}{h}\] 

Exercise 29.7 | Q 63 | Page 51

If  \[\lim_{x \to 0} kx  cosec x = \lim_{x \to 0} x  cosec kx,\] 

Exercise 29.8 [Pages 62 - 63]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.8 [Pages 62 - 63]

Exercise 29.8 | Q 1 | Page 62

\[\lim_{x \to \pi} \frac{\sin x}{\pi - x}\]

Exercise 29.8 | Q 2 | Page 62

\[\lim_{x \to \frac{\pi}{2}} \frac{\sin 2x}{\cos x}\] 

Exercise 29.8 | Q 3 | Page 62

\[\lim_{x \to \frac{\pi}{2}} \frac{\cos^2 x}{1 - \sin x}\]

Exercise 29.8 | Q 4 | Page 62

Evaluate the following limit:

\[\lim_{x \to \frac{\pi}{3}} \frac{\sqrt{1 - \cos6x}}{\sqrt{2}\left( \frac{\pi}{3} - x \right)}\]

Exercise 29.8 | Q 5 | Page 62

\[\lim_{x \to a} \frac{\cos x - \cos a}{x - a}\] 

Exercise 29.8 | Q 6 | Page 62

\[\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{x - \frac{\pi}{4}}\] 

Exercise 29.8 | Q 7 | Page 62

\[\lim_{x \to \frac{\pi}{2}} \frac{1 - \sin x}{\left( \frac{\pi}{2} - x \right)^2}\]

Exercise 29.8 | Q 8 | Page 62

\[\lim_{x \to \frac{\pi}{3}} \frac{\sqrt{3} - \tan x}{\pi - 3x}\]

Exercise 29.8 | Q 9 | Page 62

\[\lim_{x \to \frac{\pi}{2}} \frac{\cot x}{\frac{\pi}{2} - x}\]

Exercise 29.8 | Q 10 | Page 62

\[\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{\cos x} - \sqrt{\sin x}}{x - \frac{\pi}{4}}\] 

Exercise 29.8 | Q 11 | Page 62

\[\lim_{x \to \frac{\pi}{2}} \frac{\sqrt{2 - \sin x} - 1}{\left( \frac{\pi}{2} - x \right)^2}\] 

Exercise 29.8 | Q 12 | Page 62

\[\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2} - \cos x - \sin x}{\left( \frac{\pi}{4} - x \right)^2}\] 

Exercise 29.8 | Q 13 | Page 62

\[\lim_{x \to \frac{\pi}{8}} \frac{\cot 4x - \cos 4x}{\left( \pi - 8x \right)^3}\] 

Exercise 29.8 | Q 14 | Page 62

\[\lim_{x \to a} \frac{\cos x - \cos a}{\sqrt{x} - \sqrt{a}}\]

Exercise 29.8 | Q 15 | Page 62

\[\lim_{x \to \pi} \frac{\sqrt{5 + \cos x} - 2}{\left( \pi - x \right)^2}\] 

Exercise 29.8 | Q 16 | Page 62

\[\lim_{x \to a} \frac{\cos \sqrt{x} - \cos \sqrt{a}}{x - a}\] 

Exercise 29.8 | Q 17 | Page 62

\[\lim_{x \to a} \frac{\sin \sqrt{x} - \sin \sqrt{a}}{x - a}\] 

Exercise 29.8 | Q 18 | Page 62

\[\lim_{x \to 1} \frac{1 - x^2}{\sin 2\pi x}\] 

Exercise 29.8 | Q 19 | Page 62

\[\lim_{x \to \frac{\pi}{4}} \frac{f\left( x \right) - f\left( \frac{\pi}{4} \right)}{x - \frac{\pi}{4}},\]

Exercise 29.8 | Q 20 | Page 62

\[\lim_{x \to 1} \frac{1 + \cos \pi x}{\left( 1 - x \right)^2}\] 

Exercise 29.8 | Q 21 | Page 62

\[\lim_{x \to 1} \frac{1 - x^2}{\sin \pi x}\]

Exercise 29.8 | Q 22 | Page 62

\[\lim_{x \to \frac{\pi}{4}} \frac{1 - \sin 2x}{1 + \cos 4x}\] 

Exercise 29.8 | Q 23 | Page 62

\[\lim_{x \to 1} \frac{1 - \frac{1}{x}}{\sin \pi \left( x - 1 \right)}\]

Exercise 29.8 | Q 24 | Page 62
\[\lim_{n \to \infty} n \sin \left( \frac{\pi}{4 n} \right) \cos \left( \frac{\pi}{4 n} \right)\]

 

Exercise 29.8 | Q 25 | Page 62

\[\lim_{n \to \infty} 2^{n - 1} \sin \left( \frac{a}{2^n} \right)\] 

 

Exercise 29.8 | Q 26 | Page 62

\[\lim_{n \to \infty} \frac{\sin \left( \frac{a}{2^n} \right)}{\sin \left( \frac{b}{2^n} \right)}\]

Exercise 29.8 | Q 27 | Page 62

\[\lim_{x \to - 1} \frac{x^2 - x - 2}{\left( x^2 + x \right) + \sin \left( x + 1 \right)}\]

Exercise 29.8 | Q 28 | Page 62

\[\lim_{x \to 2} \frac{x^2 - x - 2}{x^2 - 2x + \sin \left( x - 2 \right)}\] 

Exercise 29.8 | Q 29 | Page 63

\[\lim_{x \to 1} \left( 1 - x \right) \tan \left( \frac{\pi x}{2} \right)\]

Exercise 29.8 | Q 30 | Page 63

\[\lim_{x \to \frac{\pi}{4}} \frac{1 - \tan x}{1 - \sqrt{2} \sin x}\] 

Exercise 29.8 | Q 31 | Page 63

\[\lim_{x \to \pi} \frac{\sqrt{2 + \cos x} - 1}{\left( \pi - x \right)^2}\]

Exercise 29.8 | Q 32 | Page 63

\[\lim_{x \to \pi} \frac{1 + \cos x}{\tan^2 x}\] 

Exercise 29.8 | Q 33 | Page 63

\[\lim_{x \to \frac{\pi}{2}} \left( \frac{\pi}{2} - x \right) \tan x\]

Exercise 29.8 | Q 34 | Page 63

\[\lim_{x \to \frac{\pi}{6}} \frac{\cot^2 x - 3}{cosec x - 2}\]

Exercise 29.8 | Q 35 | Page 63

\[\lim_{x \to \frac{\pi}{4}} \frac{\sqrt{2} - \cos x - \sin x}{\left( 4x - \pi \right)^2}\]

Exercise 29.8 | Q 36 | Page 63

\[\lim_{x \to \frac{\pi}{2}} \frac{\left( \frac{\pi}{2} - x \right) \sin x - 2 \cos x}{\left( \frac{\pi}{2} - x \right) + \cot x}\]

Exercise 29.8 | Q 37 | Page 63

\[\lim_{x \to \frac{\pi}{4}} \frac{\cos x - \sin x}{\left( \frac{\pi}{4} - x \right) \left( \cos x + \sin x \right)}\]

Exercise 29.8 | Q 38 | Page 63

Evaluate the following limit:

\[\lim_{x \to \pi} \frac{1 - \sin\frac{x}{2}}{\cos\frac{x}{2}\left( \cos\frac{x}{4} - \sin\frac{x}{4} \right)}\]

 

Exercise 29.9 [Page 65]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.9 [Page 65]

Exercise 29.9 | Q 1 | Page 65

\[\lim_{x \to \pi} \frac{1 + \cos x}{\tan^2 x}\] 

Exercise 29.9 | Q 2 | Page 65

\[\lim_{x \to \frac{\pi}{4}} \frac{{cosec}^2 x - 2}{\cot x - 1}\]

Exercise 29.9 | Q 3 | Page 65

\[\lim_{x \to \frac{\pi}{6}} \frac{\cot^2 x - 3}{cosec x - 2}\]

Exercise 29.9 | Q 4 | Page 65

\[\lim_{x \to \frac{\pi}{4}} \frac{2 - {cosec}^2 x}{1 - \cot x}\] 

Exercise 29.9 | Q 5 | Page 65

\[\lim_{x \to \pi} \frac{\sqrt{2 + \cos x} - 1}{\left( \pi - x \right)^2}\] 

Exercise 29.9 | Q 6 | Page 65

\[\lim_{x \to \frac{3\pi}{2}} \frac{1 + {cosec}^3 x}{\cot^2 x}\]

Exercise 29.1 [Pages 71 - 72]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.1 [Pages 71 - 72]

Exercise 29.1 | Q 1 | Page 71

\[\lim_{x \to 0} \frac{5^x - 1}{\sqrt{4 + x} - 2}\]

Exercise 29.1 | Q 2 | Page 71

\[\lim_{x \to 0} \frac{\log \left( 1 + x \right)}{3^x - 1}\]

Exercise 29.1 | Q 3 | Page 71

\[\lim_{x \to 0} \frac{a^x + a^{- x} - 2}{x^2}\]

Exercise 29.1 | Q 4 | Page 71

\[\lim_{x \to 0} \frac{a^{mx} - 1}{b^{nx} - 1}, n \neq 0\]

Exercise 29.1 | Q 5 | Page 71

\[\lim_{x \to 0} \frac{a^x + b^x - 2}{x}\]

Exercise 29.1 | Q 6 | Page 71

\[\lim_{x \to 0} \frac{9^x - 2 . 6^x + 4^x}{x^2}\] 

Exercise 29.1 | Q 7 | Page 71

\[\lim_{x \to 0} \frac{8^x - 4^x - 2^x + 1}{x^2}\]

Exercise 29.1 | Q 8 | Page 71

\[\lim_{x \to 0} \frac{a^{mx} - b^{nx}}{x}\] 

Exercise 29.1 | Q 9 | Page 71

\[\lim_{x \to 0} \frac{a^x + b^x + c^x - 3}{x}\] 

Exercise 29.1 | Q 10 | Page 71

\[\lim_{x \to 2} \frac{x - 2}{\log_a \left( x - 1 \right)}\]

Exercise 29.1 | Q 11 | Page 71

\[\lim_{x \to 0} \frac{5^x + 3^x + 2^x - 3}{x}\]

Exercise 29.1 | Q 12 | Page 71

\[\lim_{x \to \infty} \left( a^{1/x} - 1 \right)x\]

Exercise 29.1 | Q 13 | Page 71

\[\lim_{x \to 0} \frac{a^{mx} - b^{nx}}{\sin kx}\]

Exercise 29.1 | Q 14 | Page 71

\[\lim_{x \to 0} \frac{a^x + b^ x - c^x - d^x}{x}\]

Exercise 29.1 | Q 15 | Page 71

\[\lim_{x \to 0} \frac{e^x - 1 + \sin x}{x}\]

Exercise 29.1 | Q 16 | Page 71

\[\lim_{x \to 0} \frac{\sin 2x}{e^x - 1}\] 

Exercise 29.1 | Q 17 | Page 71

\[\lim_{x \to 0} \frac{e\sin x - 1}{x}\] 

Exercise 29.1 | Q 18 | Page 71

\[\lim_{x \to 0} \frac{e^{2x} - e^x}{\sin 2x}\]

Exercise 29.1 | Q 19 | Page 71

\[\lim_{x \to a} \frac{\log x - \log a}{x - a}\] 

Exercise 29.1 | Q 20 | Page 71

\[\lim_{x \to 0} \frac{\log \left( a + x \right) - \log \left( a - x \right)}{x}\]

Exercise 29.1 | Q 21 | Page 71

\[\lim_{x \to 0} \frac{\log \left( 2 + x \right) + \log 0 . 5}{x}\]

Exercise 29.1 | Q 22 | Page 71

\[\lim_{x \to 0} \frac{\log \left( a + x \right) - \log a}{x}\]

Exercise 29.1 | Q 23 | Page 71

\[\lim_{x \to 0} \frac{\log \left( 3 + x \right) - \log \left( 3 - x \right)}{x}\] 

Exercise 29.1 | Q 24 | Page 71

\[\lim_{x \to 0} \frac{8^x - 2^x}{x}\]

Exercise 29.1 | Q 25 | Page 71

\[\lim_{x \to 0} \frac{x\left( 2^x - 1 \right)}{1 - \cos x}\] 

Exercise 29.1 | Q 26 | Page 71

\[\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{\log \left( 1 + x \right)}\] 

Exercise 29.1 | Q 27 | Page 71

\[\lim_{x \to 0} \frac{\log \left| 1 + x^3 \right|}{\sin^3 x}\] 

 

Exercise 29.1 | Q 28 | Page 71

`\lim_{x \to \pi/2} \frac{a^\cot x - a^\cos x}{\cot x - \cos x}`

Exercise 29.1 | Q 29 | Page 71

\[\lim_{x \to 0} \frac{e^x - 1}{\sqrt{1 - \cos x}}\]

Exercise 29.1 | Q 30 | Page 71

\[\lim_{x \to 5} \frac{e^x - e^5}{x - 5}\]

Exercise 29.1 | Q 31 | Page 72

\[\lim_{x \to 0} \frac{e^{x + 2} - e^2}{x}\] 

Exercise 29.1 | Q 32 | Page 72

`\lim_{x \to \pi/2} \frac{e^\cos x - 1}{\cos x}`

Exercise 29.1 | Q 33 | Page 72

\[\lim_{x \to 0} \frac{e^{3 + x} - \sin x - e^3}{x}\] 

Exercise 29.1 | Q 34 | Page 72

\[\lim_{x \to 0} \frac{e^x - x - 1}{2}\] 

Exercise 29.1 | Q 35 | Page 72

\[\lim_{x \to 0} \frac{e^{3x} - e^{2x}}{x}\] 

Exercise 29.1 | Q 36 | Page 72

`\lim_{x \to 0} \frac{e^\tan x - 1}{\tan x}`

Exercise 29.1 | Q 37 | Page 72

\[\lim_{x \to 0} \frac{e^{bx} - e^{ax}}{x} \text{ where } 0 < a < b\] 

Exercise 29.1 | Q 38 | Page 72

`\lim_{x \to 0} \frac{e^\tan x - 1}{x}`

Exercise 29.1 | Q 39 | Page 72

`\lim_{x \to 0} \frac{e^x - e^\sin x}{x - \sin x}`

Exercise 29.1 | Q 40 | Page 72

\[\lim_{x \to 0} \frac{3^{2 + x} - 9}{x}\]

Exercise 29.1 | Q 41 | Page 72

\[\lim_{x \to 0} \frac{a^x - a^{- x}}{x}\]

Exercise 29.1 | Q 42 | Page 72

\[\lim_{x \to 0} \frac{x\left( e^x - 1 \right)}{1 - \cos x}\]

Exercise 29.1 | Q 43 | Page 72

\[\lim_{x \to \pi/2} \frac{2^{- \cos x} - 1}{x\left( x - \frac{\pi}{2} \right)}\]

Exercise 29.11 [Pages 76 - 77]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 LimitsExercise 29.11 [Pages 76 - 77]

Exercise 29.11 | Q 1 | Page 76

\[\lim_{n \to \infty} \left( 1 + \frac{x}{n} \right)^n\]

Exercise 29.11 | Q 2 | Page 76

\[\lim_{x \to 0^+} \left\{ 1 + \tan^2 \sqrt{x} \right\}^{1/2x}\]

Exercise 29.11 | Q 3 | Page 76

\[\lim_{x \to 0} \left( \cos x \right)^{1/\sin x}\] 

Exercise 29.11 | Q 4 | Page 76

\[\lim_{x \to 0} \left( \cos x + \sin x \right)^{1/x}\]

Exercise 29.11 | Q 5 | Page 77

\[\lim_{x \to 0} \left( \cos x + a \sin bx \right)^{1/x}\]

Exercise 29.11 | Q 6 | Page 77

\[\lim_{x \to \infty} \left\{ \frac{x^2 + 2x + 3}{2 x^2 + x + 5} \right\}^\frac{3x - 2}{3x + 2}\]

Exercise 29.11 | Q 7 | Page 77

\[\lim_{x \to 1} \left\{ \frac{x^3 + 2 x^2 + x + 1}{x^2 + 2x + 3} \right\}^\frac{1 - \cos \left( x - 1 \right)}{\left( x - 1 \right)^2}\]

Exercise 29.11 | Q 8 | Page 77

\[\lim_{x \to 0} \left\{ \frac{e^x + e^{- x} - 2}{x^2} \right\}^{1/ x^2}\]

Exercise 29.11 | Q 9 | Page 77

\[\lim_{x \to a} \left\{ \frac{\sin x}{\sin a} \right\}^\frac{1}{x - a}\]

Exercise 29.11 | Q 10 | Page 77

\[\lim_{x \to \infty} \left\{ \frac{3 x^2 + 1}{4 x^2 - 1} \right\}^\frac{x^3}{1 + x}\]

[Page 77]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 Limits [Page 77]

Q 1 | Page 77

Write the value of \[\lim_{x \to 0} \frac{\sqrt{1 - \cos 2x}}{x} .\]

Q 2 | Page 77

Write the value of \[\lim_{x \to 0^-} \left[ x \right] .\]

 
Q 3 | Page 77

Write the value of \[\lim_{x \to 0^+} \left[ x \right] .\]

Q 4 | Page 77

Write the value of \[\lim_{x \to 1^-} x - \left[ x \right] .\] 

Q 5 | Page 77

\[\lim_{x \to 0^-} \frac{\sin \left[ x \right]}{\left[ x \right]} .\] 

Q 6 | Page 77

\[\lim_{x \to \pi} \frac{\sin x}{x - \pi} .\] 

Q 7 | Page 77

Write the value of \[\lim_{x \to \infty} \frac{\sin x}{x} .\] 

Q 8 | Page 77

\[\lim_{x \to 0} \left\{ \frac{e^x + e^{- x} - 2}{x^2} \right\}^{1/ x^2}\]

Q 9 | Page 77

\[\lim_{x \to a} \left\{ \frac{\sin x}{\sin a} \right\}^\frac{1}{x - a}\]

Q 10 | Page 77

\[\lim_{x \to \infty} \left\{ \frac{3 x^2 + 1}{4 x^2 - 1} \right\}^\frac{x^3}{1 + x}\]

Q 11 | Page 77

\[\lim_{x \to 0} \frac{\sin x}{\sqrt{1 + x} - 1} .\] 

Q 12 | Page 77

Write the value of \[\lim_{x \to - \infty} \left( 3x + \sqrt{9 x^2 - x} \right) .\]

Q 13 | Page 77

Write the value of \[\lim_{n \to \infty} \frac{n! + \left( n + 1 \right)!}{\left( n + 1 \right)! + \left( n + 2 \right)!} .\]

Q 14 | Page 77

Write the value of \[\lim_{x \to \pi/2} \frac{2x - \pi}{\cos x} .\] 

Q 15 | Page 77

Write the value of \[\lim_{n \to \infty} \frac{1 + 2 + 3 + . . . + n}{n^2} .\]

[Pages 77 - 81]

RD Sharma solutions for Class 11 Mathematics Textbook Chapter 29 Limits [Pages 77 - 81]

Q 1 | Page 77

\[\lim_{n \to \infty} \frac{1^2 + 2^2 + 3^2 + . . . + n^2}{n^3}\] 

  • (a) 1

  • (b) 1/2 

  • (c) 1/3 

  • (d) 0 

Q 2 | Page 78

\[\lim_{x \to 0} \frac{\sin 2x}{x}\] 

  • (a) 0 

  • (b) 1 

  • (c) 1/2 

  • (d) 2 

Q 3 | Page 78

If \[f\left( x \right) = x \sin \left( 1/x \right), x \neq 0,\]  then \[\lim_{x \to 0} f\left( x \right) =\] 

  • (a) 1 

  • (b) 0

  • (c) −1 

  • (d) does not exist

Q 4 | Page 78

\[\lim_{x \to  } \frac{1 - \cos 2x}{x} is\]

  • (a) 0 

  • (b) 1 

  • (c) 2 

  • (d) 4 

Q 5 | Page 78

\[\lim_{x \to 0}  \frac{\left( 1 - \cos 2x \right) \sin 5x}{x^2 \sin 3x} =\]

  • (a) 10/3 

  • (b) 3/10 

  • (c) 6/5 

  • (d) 5/6

Q 6 | Page 78

\[\lim_{x \to 0} \frac{x}{\tan x} is\] 

  • (a) 0 

  • (b) 1 

  • (c) 4 

  • (d) not defined 

Q 7 | Page 78

\[\lim_{n \to \infty} \left\{ \frac{1}{1 - n^2} + \frac{2}{1 - n^2} + . . . + \frac{n}{1 - n^2} \right\}\]

  • (a) 0

  • (b) −1/2

  • (c) 1/2

  • (d) none of these 

Q 8 | Page 78

\[\lim_{x \to \infty} \frac{\sin x}{x}\] equals 

  •  0 

  •  ∞ 

  •  1

  •  does not exist 

Q 9 | Page 78

\[\lim_{x \to 0} \frac{\sin x^0}{x}\] 

  • 1

  • π

  •  π/180

Q 10 | Page 78

\[\lim_{x \to 3} \frac{x - 3}{\left| x - 3 \right|},\] is equal to

  •  1 

  • −1 

  •  0 

  • does not exist 

Q 11 | Page 78

\[\lim_{x \to a} \frac{x^n - a^n}{x - a}\]  is equal at 

  • na

  • nan−1 

  • na 

  •  1

     
Q 12 | Page 78

\[\lim_{x \to \pi/4} \frac{\sqrt{2} \cos x - 1}{\cot x - 1}\] is equal to

  • \[\frac{1}{\sqrt{2}}\] 

  • \[\frac{1}{2}\] 

  • \[\frac{1}{2\sqrt{2}}\] 

Q 13 | Page 78

\[\lim_{x \to \infty} \frac{\sqrt{x^2 - 1}}{2x + 1}\] 

  • 1

  • 0

  • −1 

  • 1/2 

Q 14 | Page 78

\[\lim 2_{h \to 0} \left\{ \frac{\sqrt{3} \sin \left( \pi/6 + h \right) - \cos \left( \pi/6 + h \right)}{\sqrt{3} h \left( \sqrt{3} \cos h - \sin h \right)} \right\}\] 

  •  2/3 

  • 4/3 

  • \[- 2\sqrt{3}\] 

  • −4/3

Q 15 | Page 79

\[\lim_{h \to 0} \left\{ \frac{1}{h\sqrt[3]{8 + h}} - \frac{1}{2h} \right\} =\]

  • −1/12 

  • −4/3 

  • −16/3

  •  −1/48

Q 16 | Page 79

\[\lim_{n \to \infty} \left\{ \frac{1}{1 . 3} + \frac{1}{3 . 5} + \frac{1}{5 . 7} + . . . + \frac{1}{\left( 2n + 1 \right) \left( 2n + 3 \right)} \right\}\]is equal to

  •  0 

  •  1/2 

  • 1/9

  • 2

Q 17 | Page 79

\[\lim_{x \to 1} \frac{\sin \pi x}{x - 1}\] 

  • π 

  • π 

  • \[- \frac{1}{\pi}\] 

  • \[\frac{1}{\pi}\] 

Q 18 | Page 79

If \[\lim_{x \to 1} \frac{x + x^2 + x^3 + . . . + x^n - n}{x - 1} = 5050\] then n equal

  • 10 

  • 100 

  • 150 

  • none of these 

Q 19 | Page 79

The value of \[\lim_{x \to \infty} \frac{\sqrt{1 + x^4} + \left( 1 + x^2 \right)}{x^2}\]  is

  • −1 

  •  1 

  • none of these 

Q 20 | Page 79

\[\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}\] is equal to 

  • \[\frac{1}{2}\] 

  • 2

  • 1

Q 21 | Page 79

\[\lim_{x \to \pi/3} \frac{\sin \left( \frac{\pi}{3} - x \right)}{2 \cos x - 1}\] is equal to 

  • \[\sqrt{3}\]

  • \[\frac{1}{2}\]

     

  • \[\frac{1}{\sqrt{3}}\]

  • \[\sqrt{3}\]

Q 22 | Page 79

\[\lim_{x \to 3} \frac{\sum^n_{r = 1} x^r - \sum^n_{r = 1} 3^r}{x - 3}\]is real to

  • \[\frac{\left( 2n - 1 \right) \times 3^n}{4}\] 

  • \[\frac{\left( 2n - 1 \right) \times 3^n + 1}{4}\]

  • \[\left( 2n - 1 \right) 3^n + 1\] 

  • \[\frac{\left( 2n - 1 \right) \times 3^n - 1}{4}\] 

Q 23 | Page 79

\[\lim_{n \to \infty} \frac{1 - 2 + 3 - 4 + 5 - 6 + . . . . + \left( 2n - 1 \right) - 2n}{\sqrt{n^2 + 1} + \sqrt{n^2 - 1}}\] is equal to 

  • \[\frac{1}{2}\]

  • \[- \frac{1}{2}\]

  •  1

  •  −1 

Q 24 | Page 79

If \[f\left( x \right) = \left\{ \begin{array}{l}x \sin \frac{1}{x}, & x \neq 0 \\ 0, & x = 0\end{array}, \right.\] then \[\lim_{x \to 0} f\left( x \right)\]  equals 

  •  1 

  •  0 

  •  −1 

  •  none of these 

Q 25 | Page 80

\[\lim_{n \to \infty} \frac{n!}{\left( n + 1 \right)! + n!}\]  is equal to

  • \[\frac{1}{2}\] 

  •  2 

Q 26 | Page 80

\[\lim_{x \to \pi/4} \frac{4\sqrt{2} - \left( \cos x + \sin x \right)^5}{1 - \sin 2x}\] is equal to 

  • \[5\sqrt{2}\] 

  • \[3\sqrt{2}\]

  • \[\sqrt{2}\] 

  •  none of these

Q 27 | Page 80

\[\lim_{x \to 2} \frac{\sqrt{1 + \sqrt{2 + x} - \sqrt{3}}}{x - 2}\] is equal to 

  • \[\frac{1}{8\sqrt{3}}\]

  • \[\frac{1}{\sqrt{3}}\]

  • $\mathnormal{8 \sqrt{3}}$ 

  • \[\sqrt{3}\]

Q 28 | Page 80

\[\lim_{x \to \infty} a^x \sin \left( \frac{b}{a^x} \right), a, b > 1\] is equal to 

  •  a

  •  a loge b

  • b loge a

Q 29 | Page 80

\[\lim_{x \to 0} \frac{8}{x^8}\left\{ 1 - \cos \frac{x^2}{2} - \cos \frac{x^2}{4} + \cos \frac{x^2}{2} \cos \frac{x^2}{4} \right\}\] is equal to 

  • \[\frac{1}{16}\] 

  • \[- \frac{1}{16}\] 

  • \[\frac{1}{32}\] 

  • \[- \frac{1}{32}\] 

Q 30 | Page 80

If α is a repeated root of ax2 + bx + c = 0, then \[\lim_{x \to \alpha} \frac{\tan \left( a x^2 + bx + c \right)}{\left( x - \alpha \right)^2}\]

  •  

  •  

  •  0

Q 31 | Page 80

The value of \[\lim_{x \to 0} \frac{\sqrt{a^2 - ax + x^2} - \sqrt{a^2 + ax + x^2}}{\sqrt{a + x} - \sqrt{a - x}}\] 

  • \[\sqrt{a}\] 

  • \[- \sqrt{a}\]

Q 32 | Page 80

The value of \[\lim_{x \to 0} \frac{1 - \cos x + 2 \sin x - \sin^3 x - x^2 + 3 x^4}{\tan^3 x - 6 \sin^2 x + x - 5 x^3}\] is 

  • 1

  • −1 

  • −2

Q 33 | Page 80

\[\lim_\theta \to \pi/2 \frac{1 - \sin \theta}{\left( \pi/2 - \theta \right) \cos \theta}\] is equal to 

  •  1

  • −1 

  • \[\frac{1}{2}\]

  • \[- \frac{1}{2}\] 

Q 34 | Page 80

The value of \[\lim_{x \to \pi/2} \left( \sec x - \tan x \right)\]is 

  • 2

  • −1

  •  1

  • 0

Q 35 | Page 80

The value of \[\lim_{x \to \infty} \frac{n!}{\left( n + 1 \right)! - n!}\] 

  •  1 

  •  −1 

  • none of these 

Q 36 | Page 80

The value of \[\lim_{n \to \infty} \frac{\left( n + 2 \right)! + \left( n + 1 \right)!}{\left( n + 2 \right)! - \left( n + 1 \right)!}\] is 

  •  0 

  • −1 

  •  1 

  • none of these

Q 37 | Page 80

The value of \[\lim_{x \to \infty} \frac{\left( x + 1 \right)^{10} + \left( x + 2 \right)^{10} + . . . + \left( x + 100 \right)^{10}}{x^{10} + {10}^{10}}\] is 

  • 10 

  •  100 

  • 1010 

  • none of these

     

Q 38 | Page 81

The value of \[\lim_{n \to \infty} \left\{ \frac{1 + 2 + 3 + . . . + n}{n + 2} - \frac{n}{2} \right\}\] 

  • 1/2

  • −1

  • −1/2 

Q 39 | Page 81

\[\lim_{x \to 1} \left[ x - 1 \right]\] where [.] is the greatest integer function, is equal to 

  •  1  

  • 2    

  •  0  

  • does not exist                                

Q 40 | Page 81

\[\lim_{x \to \infty} \frac{\left| x \right|}{x}\]  is equal to 

  •  1     

  • −1         

  •  0     

  •  does not exist 

Q 41 | Page 81

\[\lim_{x \to 0} \frac{\left| \sin x \right|}{x}\]

  • 1          

  • −1       

  • 0             

  • does not exist 

Q 42 | Page 81

If \[f\left( x \right) = \begin{cases}\frac{\sin\left[ x \right]}{\left[ x \right]}, & \left[ x \right] \neq 0 \\ 0, & \left[ x \right] = 0\end{cases}\]  where  denotes the greatest integer function, then \[\lim_{x \to 0} f\left( x \right)\]  

  • 1  

  • 0  

  • −1     

  • does not exist                                    

Chapter 29: Limits

Exercise 29.1Exercise 29.2Exercise 29.3Exercise 29.4Exercise 29.5Exercise 29.6Exercise 29.7Exercise 29.8Exercise 29.9Exercise 29.11Others
Class 11 Mathematics Textbook - Shaalaa.com

RD Sharma solutions for Class 11 Mathematics Textbook chapter 29 - Limits

RD Sharma solutions for Class 11 Mathematics Textbook chapter 29 (Limits) include all questions with solution and detail explanation. This will clear students doubts about any question and improve application skills while preparing for board exams. The detailed, step-by-step solutions will help you understand the concepts better and clear your confusions, if any. Shaalaa.com has the CBSE Class 11 Mathematics Textbook solutions in a manner that help students grasp basic concepts better and faster.

Further, we at Shaalaa.com provide such solutions so that students can prepare for written exams. RD Sharma textbook solutions can be a core help for self-study and acts as a perfect self-help guidance for students.

Concepts covered in Class 11 Mathematics Textbook chapter 29 Limits are Limits of Exponential Functions, Derivative of Slope of Tangent of the Curve, Theorem for Any Positive Integer n, Graphical Interpretation of Derivative, Derive Derivation of x^n, Algebra of Derivative of Functions, Derivative of Polynomials and Trigonometric Functions, Derivative Introduced as Rate of Change Both as that of Distance Function and Geometrically, Limits of Logarithmic Functions, Intuitive Idea of Derivatives, Introduction of Limits, Introduction to Calculus, Algebra of Limits, Limits of Polynomials and Rational Functions, Introduction of Derivatives, Limits of Trigonometric Functions.

Using RD Sharma Class 11 solutions Limits exercise by students are an easy way to prepare for the exams, as they involve solutions arranged chapter-wise also page wise. The questions involved in RD Sharma Solutions are important questions that can be asked in the final exam. Maximum students of CBSE Class 11 prefer RD Sharma Textbook Solutions to score more in exam.

Get the free view of chapter 29 Limits Class 11 extra questions for Class 11 Mathematics Textbook and can use Shaalaa.com to keep it handy for your exam preparation

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