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

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.

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.

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. 

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

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 \[\lim_{x \to 0} f\left( x \right)\] 

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 \[\lim_{x \to 1} f\left( x \right)\]

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

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

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

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

Evaluate the following one sided limit: 

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

Evaluate the following one sided limit: 

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

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:

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