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

## Mathematics Class 11

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## Chapter 29: Limits

Ex. 29.10Ex. 29.20Ex. 29.30Ex. 29.40Ex. 29.50Ex. 29.60Ex. 29.70Ex. 29.80Ex. 29.90Ex. 29.11Others

#### Chapter 29: Limits Exercise 29.10 solutions [Pages 11 - 23]

Ex. 29.10 | Q 1 | Page 11

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

Ex. 29.10 | Q 2 | Page 11

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

Ex. 29.10 | Q 3 | Page 11

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

Ex. 29.10 | 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.

Ex. 29.10 | 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.

Ex. 29.10 | 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.

Ex. 29.10 | Q 7 | Page 11

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

Ex. 29.10 | 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 $\lim_{x \to 0} f\left( x \right)$

Ex. 29.10 | 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 $\lim_{x \to 1} f\left( x \right)$

Ex. 29.10 | Q 9 | Page 11

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

Ex. 29.10 | Q 10 | Page 11

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

Ex. 29.10 | 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 $\lim_{x \to a} f\left( x \right) .$

Ex. 29.10 | Q 12 | Page 11

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

Ex. 29.10 | Q 13.01 | Page 11

Evaluate the following one sided limit:

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

Ex. 29.10 | Q 13.02 | Page 11

Evaluate the following one sided limit:

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

Ex. 29.10 | Q 13.03 | Page 11

Evaluate the following one sided limit:

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

Ex. 29.10 | Q 13.04 | Page 11

Evaluate the following one sided limit:

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

Ex. 29.10 | Q 13.05 | Page 11

Evaluate the following one sided limit:

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

Ex. 29.10 | Q 13.06 | Page 11

Evaluate the following one sided limit:

$\lim_{x \to \frac{\pi}{2}} \tan x$

Ex. 29.10 | Q 13.07 | Page 11

Evaluate the following one sided limit:

$\lim_{x \to \frac{\pi}{2}} \tan x$

Ex. 29.10 | Q 13.08 | Page 11

Evaluate the following one sided limit:

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

Ex. 29.10 | Q 13.09 | Page 11

Evaluate the following one sided limit:

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

Ex. 29.10 | Q 13.1 | Page 11

Evaluate the following one sided limit:

$\lim_{x \to 0^-} 2 - \cot x$

Ex. 29.10 | Q 13.11 | Page 11

Evaluate the following one sided limit:

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

Ex. 29.10 | Q 14 | Page 12

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

Ex. 29.10 | Q 15.1 | Page 12

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

Ex. 29.10 | Q 15.2 | Page 12

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

Ex. 29.10 | Q 15.3 | Page 12

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

Ex. 29.10 | Q 16 | Page 12

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

Ex. 29.10 | Q 17 | Page 12

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

Ex. 29.10 | Q 18 | Page 12

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

Ex. 29.10 | Q 19 | Page 12

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

Ex. 29.10 | Q 20 | Page 12

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

Ex. 29.10 | Q 21 | Page 23

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

Ex. 29.10 | Q 21 | Page 12

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

Ex. 29.10 | 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 $\lim_{x \to \frac{\pi}{2}} f\left( x \right) = f\left( \frac{\pi}{2} \right)$

#### Chapter 29: Limits Exercise 29.20 solutions [Page 18]

Ex. 29.20 | Q 1 | Page 18

$\lim_{x \to 1} \frac{x^2 + 1}{x + 1}$

Ex. 29.20 | Q 2 | Page 18

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

Ex. 29.20 | Q 3 | Page 18

$\lim_{x \to 3} \frac{\sqrt{2x + 3}}{x + 3}$

Ex. 29.20 | Q 4 | Page 18

$\lim_{x \to 1} \frac{\sqrt{x + 8}}{\sqrt{x}}$

Ex. 29.20 | Q 5 | Page 18

$\lim_{x \to a} \frac{\sqrt{x} + \sqrt{a}}{x + a}$

Ex. 29.20 | Q 6 | Page 18

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

Ex. 29.20 | Q 7 | Page 18

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

Ex. 29.20 | Q 8 | Page 18

$\lim_{x \to 0} 9$

Ex. 29.20 | Q 9 | Page 18

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

Ex. 29.20 | Q 10 | Page 18

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

Ex. 29.20 | Q 11 | Page 18

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

Ex. 29.20 | Q 12 | Page 18

$\lim_{x \to 0} \frac{3x + 1}{x + 3}$

Ex. 29.20 | Q 13 | Page 18

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

Ex. 29.20 | Q 14 | Page 18

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

#### Chapter 29: Limits Exercise 29.30 solutions [Pages 23 - 24]

Ex. 29.30 | Q 1 | Page 23

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

Ex. 29.30 | Q 2 | Page 23

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

Ex. 29.30 | Q 3 | Page 23

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

Ex. 29.30 | Q 4 | Page 23

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

Ex. 29.30 | Q 5 | Page 23

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

Ex. 29.30 | Q 6 | Page 23

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

Ex. 29.30 | Q 7 | Page 23

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

Ex. 29.30 | Q 8 | Page 23

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

Ex. 29.30 | Q 9 | Page 23

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

Ex. 29.30 | Q 10 | Page 23

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

Ex. 29.30 | Q 11 | Page 23

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

Ex. 29.30 | Q 12 | Page 23

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

Ex. 29.30 | Q 13 | Page 23

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

Ex. 29.30 | Q 14 | Page 23

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

Ex. 29.30 | Q 15 | Page 23

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

Ex. 29.30 | Q 16 | Page 23

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

Ex. 29.30 | Q 17 | Page 23

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

Ex. 29.30 | Q 18 | Page 23

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

Ex. 29.30 | Q 19 | Page 23

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

Ex. 29.30 | Q 20 | Page 23

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

Ex. 29.30 | Q 21 | Page 23
$\lim_{x \to 3} \left( \frac{1}{x - 3} - \frac{3}{x^2 - 3x} \right)$
Ex. 29.30 | Q 22 | Page 23

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

Ex. 29.30 | Q 23 | Page 23

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

Ex. 29.30 | Q 24 | Page 23

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

Ex. 29.30 | Q 25 | Page 23

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

Ex. 29.30 | Q 26 | Page 23

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

Ex. 29.30 | Q 27 | Page 23

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

Ex. 29.30 | Q 28 | Page 23

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

Ex. 29.30 | Q 29 | Page 23

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

Ex. 29.30 | Q 30 | Page 23

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

Ex. 29.30 | Q 31 | Page 24

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

Ex. 29.30 | Q 32 | Page 24

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

Ex. 29.30 | Q 33 | Page 24

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

Ex. 29.30 | Q 34 | Page 24

Evaluate the following limit:

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

#### Chapter 29: Limits Exercise 29.40 solutions [Pages 28 - 30]

Ex. 29.40 | Q 1 | Page 28

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

Ex. 29.40 | Q 2 | Page 28

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

Ex. 29.40 | Q 3 | Page 28

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

Ex. 29.40 | Q 4 | Page 28

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

Ex. 29.40 | Q 5 | Page 28

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

Ex. 29.40 | Q 6 | Page 28

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

Ex. 29.40 | Q 7 | Page 28

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

Ex. 29.40 | Q 8 | Page 28

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

Ex. 29.40 | Q 9 | Page 28

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

Ex. 29.40 | Q 10 | Page 28

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

Ex. 29.40 | Q 11 | Page 28

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

Ex. 29.40 | Q 12 | Page 28

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

Ex. 29.40 | Q 13 | Page 28

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

Ex. 29.40 | Q 14 | Page 28

$\lim_{x \to 2} \frac{x - 2}{\sqrt{x} - \sqrt{2}}$

Ex. 29.40 | Q 15 | Page 28

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

Ex. 29.40 | Q 16 | Page 28

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

Ex. 29.40 | Q 17 | Page 28

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

Ex. 29.40 | Q 18 | Page 28

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

Ex. 29.40 | Q 19 | Page 28

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

Ex. 29.40 | Q 20 | Page 28

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

Ex. 29.40 | Q 21 | Page 29

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

Ex. 29.40 | Q 22 | Page 29

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

Ex. 29.40 | Q 23 | Page 29

$\lim_{x \to 4} \frac{2 - \sqrt{x}}{4 - x}$

Ex. 29.40 | Q 24 | Page 29

$\lim_{x \to a} \frac{x - a}{\sqrt{x} - \sqrt{a}}$

Ex. 29.40 | Q 25 | Page 29

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

Ex. 29.40 | Q 26 | Page 29

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

Ex. 29.40 | Q 27 | Page 29

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

Ex. 29.40 | Q 28 | Page 29

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

Ex. 29.40 | Q 29 | Page 29

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

Ex. 29.40 | Q 29 | Page 30

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

Ex. 29.40 | Q 31 | Page 29

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

Ex. 29.40 | Q 32 | Page 29

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

Ex. 29.40 | Q 33 | Page 29

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

Ex. 29.40 | Q 34 | Page 29

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

#### Chapter 29: Limits Exercise 29.50 solutions [Page 33]

Ex. 29.50 | Q 1 | Page 33

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

Ex. 29.50 | Q 2 | Page 33

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

Ex. 29.50 | Q 3 | Page 33

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

Ex. 29.50 | Q 4 | Page 33

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

Ex. 29.50 | Q 5 | Page 33

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

Ex. 29.50 | Q 6 | Page 33

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

Ex. 29.50 | Q 7 | Page 33

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

Ex. 29.50 | Q 8 | Page 33

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

Ex. 29.50 | Q 9 | Page 33

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

Ex. 29.50 | Q 10 | Page 33

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

Ex. 29.50 | Q 11 | Page 33

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

Ex. 29.50 | Q 12 | Page 33

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

Ex. 29.50 | Q 13 | Page 33

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

Ex. 29.50 | Q 14 | Page 33

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

Ex. 29.50 | 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

Ex. 29.50 | 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

#### Chapter 29: Limits Exercise 29.60 solutions [Pages 38 - 39]

Ex. 29.60 | Q 1 | Page 38

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

Ex. 29.60 | Q 2 | Page 38

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

Ex. 29.60 | Q 3 | Page 38

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

Ex. 29.60 | Q 4 | Page 38

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

Ex. 29.60 | Q 5 | Page 38

$\lim_{x \to \infty} \sqrt{x + 1} - \sqrt{x}$

Ex. 29.60 | Q 6 | Page 38

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

Ex. 29.60 | Q 7 | Page 38

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

Ex. 29.60 | Q 8 | Page 38

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

Ex. 29.60 | Q 9 | Page 39

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

Ex. 29.60 | Q 10 | Page 39

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

Ex. 29.60 | Q 11 | Page 39

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

Ex. 29.60 | Q 12 | Page 39

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

Ex. 29.60 | Q 13 | Page 39

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

Ex. 29.60 | Q 14 | Page 39

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

Ex. 29.60 | Q 15 | Page 39

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

Ex. 29.60 | Q 16 | Page 39

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

Ex. 29.60 | Q 17 | Page 39

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

Ex. 29.60 | Q 18 | Page 39

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

Ex. 29.60 | Q 19 | Page 39

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

Ex. 29.60 | 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.

Ex. 29.60 | 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

Ex. 29.60 | Q 22 | Page 39

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

Ex. 29.60 | Q 23 | Page 39

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

Ex. 29.60 | Q 24 | Page 39

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

Ex. 29.60 | Q 25 | Page 39

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

Ex. 29.60 | Q 26 | Page 39

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

#### Chapter 29: Limits Exercise 29.70 solutions [Pages 49 - 51]

Ex. 29.70 | Q 1 | Page 49

$\lim_{x \to 0} \frac{\sin 3x}{5x}$

Ex. 29.70 | Q 2 | Page 49

$\lim_{x \to 0} \frac{\sin x^0}{x}$

Ex. 29.70 | Q 3 | Page 49

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

Ex. 29.70 | Q 4 | Page 49

$\lim_{x \to 0} \frac{\sin x \cos x}{3x}$

Ex. 29.70 | Q 5 | Page 50

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

Ex. 29.70 | Q 6 | Page 50

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

Ex. 29.70 | Q 7 | Page 50

$\lim_{x \to 0} \frac{\tan mx}{\tan nx}$

Ex. 29.70 | Q 8 | Page 50

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

Ex. 29.70 | Q 9 | Page 50

$\lim_{x \to 0} \frac{\sin x^n}{x^n}$

Ex. 29.70 | Q 10 | Page 50

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

Ex. 29.70 | Q 11 | Page 50

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

Ex. 29.70 | Q 12 | Page 50

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

Ex. 29.70 | Q 13 | Page 50

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

Ex. 29.70 | Q 14 | Page 50

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

Ex. 29.70 | Q 15 | Page 50

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

Ex. 29.70 | Q 16 | Page 50

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

Ex. 29.70 | Q 17 | Page 50

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

Ex. 29.70 | Q 18 | Page 50

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

Ex. 29.70 | Q 19 | Page 50

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

Ex. 29.70 | Q 20 | Page 50

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

Ex. 29.70 | Q 21 | Page 50

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

Ex. 29.70 | Q 22 | Page 50

$\lim_{x \to 0} \frac{\sin 3x - \sin x}{\sin x}$

Ex. 29.70 | Q 23 | Page 50

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

Ex. 29.70 | Q 24 | Page 50
$\lim_{x \to 0} \frac{\cos 3x - \cos 5x}{x^2}$
Ex. 29.70 | Q 25 | Page 50

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

Ex. 29.70 | Q 26 | Page 50

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

Ex. 29.70 | Q 27 | Page 50

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

Ex. 29.70 | Q 28 | Page 50

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

Ex. 29.70 | Q 29 | Page 50

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

Ex. 29.70 | Q 30 | Page 50

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

Ex. 29.70 | Q 31 | Page 50

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

Ex. 29.70 | Q 32 | Page 50

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

Ex. 29.70 | Q 33 | Page 50

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

Ex. 29.70 | Q 34 | Page 50

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

Ex. 29.70 | Q 35 | Page 50

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

Ex. 29.70 | Q 36 | Page 50

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

Ex. 29.70 | Q 37 | Page 50

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

Ex. 29.70 | Q 38 | Page 50

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

Ex. 29.70 | Q 39 | Page 50

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

Ex. 29.70 | Q 40 | Page 50

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

Ex. 29.70 | Q 41 | Page 50

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

Ex. 29.70 | Q 42 | Page 50

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

Ex. 29.70 | Q 43 | Page 51

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

Ex. 29.70 | Q 44 | Page 51

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

Ex. 29.70 | Q 45 | Page 51

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

Ex. 29.70 | Q 46 | Page 51

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

Ex. 29.70 | Q 47 | Page 51

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

Ex. 29.70 | Q 48 | Page 51

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

Ex. 29.70 | Q 49 | Page 51

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

Ex. 29.70 | Q 50 | Page 51

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

Ex. 29.70 | Q 51 | Page 51

Evaluate the following limits:

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

Ex. 29.70 | Q 52 | Page 51

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

Ex. 29.70 | Q 53 | Page 51

$\lim_{x \to 0} \frac{cosec x - \cot x}{x}$

Ex. 29.70 | Q 54 | Page 51

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

Ex. 29.70 | Q 55 | Page 51

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

Ex. 29.70 | Q 56 | Page 51

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

Ex. 29.70 | Q 57 | Page 51

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

Ex. 29.70 | Q 58 | Page 51

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

Ex. 29.70 | Q 59 | Page 51

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

Ex. 29.70 | Q 60 | Page 51

Evaluate the following limit:

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

Ex. 29.70 | Q 61 | Page 51

Evaluate the following limits:

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

Ex. 29.70 | Q 62 | Page 51

Evaluate the following limit:

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

Ex. 29.70 | Q 63 | Page 51

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

#### Chapter 29: Limits Exercise 29.80 solutions [Pages 62 - 63]

Ex. 29.80 | Q 1 | Page 62

$\lim_{x \to \pi} \frac{\sin x}{\pi - x}$

Ex. 29.80 | Q 2 | Page 62

$\lim_{x \to \frac{\pi}{2}} \frac{\sin 2x}{\cos x}$

Ex. 29.80 | Q 3 | Page 62

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

Ex. 29.80 | Q 4 | Page 62

Evaluate the following limit:

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

Ex. 29.80 | Q 5 | Page 62

$\lim_{x \to a} \frac{\cos x - \cos a}{x - a}$

Ex. 29.80 | Q 6 | Page 62

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

Ex. 29.80 | Q 7 | Page 62

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

Ex. 29.80 | Q 8 | Page 62

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

Ex. 29.80 | Q 9 | Page 62

$\lim_{x \to \frac{\pi}{2}} \frac{\cot x}{\frac{\pi}{2} - x}$

Ex. 29.80 | Q 10 | Page 62

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

Ex. 29.80 | Q 11 | Page 62

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

Ex. 29.80 | Q 12 | Page 62

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

Ex. 29.80 | Q 13 | Page 62

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

Ex. 29.80 | Q 14 | Page 62

$\lim_{x \to a} \frac{\cos x - \cos a}{\sqrt{x} - \sqrt{a}}$

Ex. 29.80 | Q 15 | Page 62

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

Ex. 29.80 | Q 16 | Page 62

$\lim_{x \to a} \frac{\cos \sqrt{x} - \cos \sqrt{a}}{x - a}$

Ex. 29.80 | Q 17 | Page 62

$\lim_{x \to a} \frac{\sin \sqrt{x} - \sin \sqrt{a}}{x - a}$

Ex. 29.80 | Q 18 | Page 62

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

Ex. 29.80 | Q 19 | Page 62

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

Ex. 29.80 | Q 20 | Page 62

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

Ex. 29.80 | Q 21 | Page 62

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

Ex. 29.80 | Q 22 | Page 62

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

Ex. 29.80 | Q 23 | Page 62

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

Ex. 29.80 | Q 24 | Page 62
$\lim_{n \to \infty} n \sin \left( \frac{\pi}{4 n} \right) \cos \left( \frac{\pi}{4 n} \right)$

Ex. 29.80 | Q 25 | Page 62

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

Ex. 29.80 | Q 26 | Page 62

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

Ex. 29.80 | Q 27 | Page 62

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

Ex. 29.80 | Q 28 | Page 62

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

Ex. 29.80 | Q 29 | Page 63

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

Ex. 29.80 | Q 30 | Page 63

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

Ex. 29.80 | Q 31 | Page 63

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

Ex. 29.80 | Q 32 | Page 63

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

Ex. 29.80 | Q 33 | Page 63

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

Ex. 29.80 | Q 34 | Page 63

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

Ex. 29.80 | Q 35 | Page 63

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

Ex. 29.80 | Q 36 | Page 63

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

Ex. 29.80 | Q 37 | Page 63

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

Ex. 29.80 | Q 38 | Page 63

Evaluate the following limit:

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

#### Chapter 29: Limits Exercise 29.90 solutions [Page 65]

Ex. 29.90 | Q 1 | Page 65

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

Ex. 29.90 | Q 2 | Page 65

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

Ex. 29.90 | Q 3 | Page 65

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

Ex. 29.90 | Q 4 | Page 65

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

Ex. 29.90 | Q 5 | Page 65

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

Ex. 29.90 | Q 6 | Page 65

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

#### Chapter 29: Limits Exercise 29.10 solutions [Pages 71 - 72]

Ex. 29.10 | Q 1 | Page 71

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

Ex. 29.10 | Q 2 | Page 71

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

Ex. 29.10 | Q 3 | Page 71

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

Ex. 29.10 | Q 4 | Page 71

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

Ex. 29.10 | Q 5 | Page 71

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

Ex. 29.10 | Q 6 | Page 71

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

Ex. 29.10 | Q 7 | Page 71

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

Ex. 29.10 | Q 8 | Page 71

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

Ex. 29.10 | Q 9 | Page 71

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

Ex. 29.10 | Q 10 | Page 71

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

Ex. 29.10 | Q 11 | Page 71

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

Ex. 29.10 | Q 12 | Page 71

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

Ex. 29.10 | Q 13 | Page 71

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

Ex. 29.10 | Q 14 | Page 71

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

Ex. 29.10 | Q 15 | Page 71

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

Ex. 29.10 | Q 16 | Page 71

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

Ex. 29.10 | Q 17 | Page 71

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

Ex. 29.10 | Q 18 | Page 71

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

Ex. 29.10 | Q 19 | Page 71

$\lim_{x \to a} \frac{\log x - \log a}{x - a}$

Ex. 29.10 | Q 20 | Page 71

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

Ex. 29.10 | Q 21 | Page 71

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

Ex. 29.10 | Q 22 | Page 71

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

Ex. 29.10 | Q 23 | Page 71

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

Ex. 29.10 | Q 24 | Page 71

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

Ex. 29.10 | Q 25 | Page 71

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

Ex. 29.10 | Q 26 | Page 71

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

Ex. 29.10 | Q 27 | Page 71

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

Ex. 29.10 | Q 28 | Page 71

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

Ex. 29.10 | Q 29 | Page 71

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

Ex. 29.10 | Q 30 | Page 71

$\lim_{x \to 5} \frac{e^x - e^5}{x - 5}$

Ex. 29.10 | Q 31 | Page 72

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

Ex. 29.10 | Q 32 | Page 72

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

Ex. 29.10 | Q 33 | Page 72

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

Ex. 29.10 | Q 34 | Page 72

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

Ex. 29.10 | Q 35 | Page 72

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

Ex. 29.10 | Q 36 | Page 72

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

Ex. 29.10 | Q 37 | Page 72

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

Ex. 29.10 | Q 38 | Page 72

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

Ex. 29.10 | Q 39 | Page 72

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

Ex. 29.10 | Q 40 | Page 72

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

Ex. 29.10 | Q 41 | Page 72

$\lim_{x \to 0} \frac{a^x - a^{- x}}{x}$

Ex. 29.10 | Q 42 | Page 72

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

Ex. 29.10 | Q 43 | Page 72

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

#### Chapter 29: Limits Exercise 29.11 solutions [Pages 76 - 77]

Ex. 29.11 | Q 1 | Page 76

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

Ex. 29.11 | Q 2 | Page 76

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

Ex. 29.11 | Q 3 | Page 76

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

Ex. 29.11 | Q 4 | Page 76

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

Ex. 29.11 | Q 5 | Page 77

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

Ex. 29.11 | Q 6 | Page 77

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

Ex. 29.11 | Q 7 | Page 77

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

Ex. 29.11 | Q 8 | Page 77

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

Ex. 29.11 | Q 9 | Page 77

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

Ex. 29.11 | Q 10 | Page 77

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

#### Chapter 29: Limits solutions [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 $\lim_{n \to \infty} \frac{1 + 2 + 3 + . . . + n}{n^2} .$

#### Chapter 29: Limits solutions [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

Ex. 29.10Ex. 29.20Ex. 29.30Ex. 29.40Ex. 29.50Ex. 29.60Ex. 29.70Ex. 29.80Ex. 29.90Ex. 29.11Others

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

RD Sharma solutions for Class 11 Maths 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 Mathematics Class 11 solutions in a manner that help students grasp basic concepts better and faster.

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

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