PUC Science इयत्ता ११ - Karnataka Board PUC Question Bank Solutions for Mathematics

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:

\[\lim_{x \to 0^-} 2 - \cot x\] 

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