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

Evaluate the following one sided limit:

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

Evaluate the following one sided limit:

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

Evaluate the following one sided limit:

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

Evaluate the following one sided limit: 

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

Evaluate the following one sided limit:

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

Evaluate the following one sided limit:

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

Evaluate the following one sided limit:

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

Evaluate the following one sided limit:

\[\lim_{x \to 0^-} 2 - \cot x\] 

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