Commerce (English Medium) कक्षा ११ - CBSE Question Bank Solutions

Evaluate the following one sided limit:

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

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

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

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

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

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

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

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

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

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

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

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 \[\lim_{x \to \frac{\pi}{2}} f\left( x \right) = f\left( \frac{\pi}{2} \right)\] 

Classify the following pair of line as coincident, parallel or intersecting:

 2x + y − 1 = 0 and 3x + 2y + 5 = 0

Classify the following pair of line as coincident, parallel or intersecting:

x − y = 0 and 3x − 3y + 5 = 0]

Classify the following pair of line as coincident, parallel or intersecting:

3x + 2y − 4 = 0 and 6x + 4y − 8 = 0.

Prove that the lines \[\sqrt{3}x + y = 0, \sqrt{3}y + x = 0, \sqrt{3}x + y = 1 \text { and } \sqrt{3}y + x = 1\]  form a rhombus.