RD Sharma solutions for Mathematics [English] Class 11 chapter 29 - Limits [Latest edition]

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

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

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

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:

\[\lim_{x \to 0^-} 2 - \cot x\] 

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. 

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

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

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

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

\[\lim_{x \to 3} \frac{\sqrt{2x + 3}}{x + 3}\] 

\[\lim_{x \to 1} \frac{\sqrt{x + 8}}{\sqrt{x}}\] 

\[\lim_{x \to a} \frac{\sqrt{x} + \sqrt{a}}{x + a}\] 

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

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

\[\lim_{x \to 0} 9\] 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Evaluate the following limit:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[\lim_{x \to 2} \frac{x - 2}{\sqrt{x} - \sqrt{2}}\] 

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

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

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

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

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

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

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

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

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

\[\lim_{x \to a} \frac{x - a}{\sqrt{x} - \sqrt{a}}\]

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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. 

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. 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[\lim_{x \to \infty} \left[ \frac{x^4 + 7 x^3 + 46x + a}{x^4 + 6} \right]\] where a is a non-zero real number. 

\[f\left( x \right) = \frac{a x^2 + b}{x^2 + 1}, \lim_{x \to 0} f\left( x \right) = 1\] and \[\lim_{x \to \infty} f\left( x \right) = 1,\]then prove that f(−2) = f(2) = 1

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

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

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

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

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

\[\lim_{x \to 0} \frac{\sin 3x}{5x}\] 

\[\lim_{x \to 0} \frac{\sin x^0}{x}\] 

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

\[\lim_{x \to 0} \frac{\sin x \cos x}{3x}\] 

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

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

\[\lim_{x \to 0} \frac{\tan mx}{\tan nx}\] 

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

\[\lim_{x \to 0} \frac{\sin x^n}{x^n}\] 

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

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

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

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

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

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

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

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

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

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

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

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

\[\lim_{x \to 0} \frac{\sin 3x - \sin x}{\sin x}\] 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Evaluate the following limits: 

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

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

\[\lim_{x \to 0} \frac{cosec x - \cot x}{x}\]

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

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

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

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

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

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

Evaluate the following limit: 

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

Evaluate the following limits: 

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

Evaluate the following limit: 

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

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

\[\lim_{x \to \pi} \frac{\sin x}{\pi - x}\]

\[\lim_{x \to \frac{\pi}{2}} \frac{\sin 2x}{\cos x}\] 

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

Evaluate the following limit:

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

\[\lim_{x \to a} \frac{\cos x - \cos a}{x - a}\] 

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

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

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

\[\lim_{x \to \frac{\pi}{2}} \frac{\cot x}{\frac{\pi}{2} - x}\]

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

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

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

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

\[\lim_{x \to a} \frac{\cos x - \cos a}{\sqrt{x} - \sqrt{a}}\]

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

\[\lim_{x \to a} \frac{\cos \sqrt{x} - \cos \sqrt{a}}{x - a}\] 

\[\lim_{x \to a} \frac{\sin \sqrt{x} - \sin \sqrt{a}}{x - a}\] 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Evaluate the following limit:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

\[\lim_{x \to a} \frac{\log x - \log a}{x - a}\] 

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

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

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

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

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

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

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

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

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

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

\[\lim_{x \to 5} \frac{e^x - e^5}{x - 5}\]

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

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

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

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

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

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

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

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

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

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

\[\lim_{x \to 0} \frac{a^x - a^{- x}}{x}\]

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

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

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

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

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

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

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

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

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

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

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

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

Write the value of \[\lim_{x \to 0} \frac{\sqrt{1 - \cos 2x}}{x} .\]

Write the value of \[\lim_{x \to 0^-} \left[ x \right] .\]

Write the value of \[\lim_{x \to 0^+} \left[ x \right] .\]

Write the value of \[\lim_{x \to 1^-} x - \left[ x \right] .\] 

\[\lim_{x \to 0^-} \frac{\sin \left[ x \right]}{\left[ x \right]} .\] 

\[\lim_{x \to \pi} \frac{\sin x}{x - \pi} .\] 

Write the value of \[\lim_{x \to \infty} \frac{\sin x}{x} .\] 

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

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

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

\[\lim_{x \to 0} \frac{\sin x}{\sqrt{1 + x} - 1} .\] 

Write the value of \[\lim_{x \to - \infty} \left( 3x + \sqrt{9 x^2 - x} \right) .\]

Write the value of \[\lim_{n \to \infty} \frac{n! + \left( n + 1 \right)!}{\left( n + 1 \right)! + \left( n + 2 \right)!} .\]

Write the value of \[\lim_{x \to \pi/2} \frac{2x - \pi}{\cos x} .\] 

Write the value of \[\lim_{n \to \infty} \frac{1 + 2 + 3 + . . . + n}{n^2} .\]

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

\[\lim_{x \to 0} \frac{\sin 2x}{x}\] 

If \[f\left( x \right) = x \sin \left( 1/x \right), x \neq 0,\]  then \[\lim_{x \to 0} f\left( x \right) =\] 

\[\lim_{x \to  } \frac{1 - \cos 2x}{x} is\]

\[\lim_{x \to 0}  \frac{\left( 1 - \cos 2x \right) \sin 5x}{x^2 \sin 3x} =\]

\[\lim_{x \to 0} \frac{x}{\tan x} is\] 

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

\[\lim_{x \to \infty} \frac{\sin x}{x}\] equals 

\[\lim_{x \to 0} \frac{\sin x^0}{x}\] 

\[\lim_{x \to 3} \frac{x - 3}{\left| x - 3 \right|},\] is equal to

\[\lim_{x \to a} \frac{x^n - a^n}{x - a}\]  is equal at 

\[\lim_{x \to \pi/4} \frac{\sqrt{2} \cos x - 1}{\cot x - 1}\] is equal to

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

\[\lim 2_{h \to 0} \left\{ \frac{\sqrt{3} \sin \left( \pi/6 + h \right) - \cos \left( \pi/6 + h \right)}{\sqrt{3} h \left( \sqrt{3} \cos h - \sin h \right)} \right\}\] 

\[\lim_{h \to 0} \left\{ \frac{1}{h\sqrt[3]{8 + h}} - \frac{1}{2h} \right\} =\]

\[\lim_{n \to \infty} \left\{ \frac{1}{1 . 3} + \frac{1}{3 . 5} + \frac{1}{5 . 7} + . . . + \frac{1}{\left( 2n + 1 \right) \left( 2n + 3 \right)} \right\}\]is equal to

\[\lim_{x \to 1} \frac{\sin \pi x}{x - 1}\] 

If \[\lim_{x \to 1} \frac{x + x^2 + x^3 + . . . + x^n - n}{x - 1} = 5050\] then n equal

The value of \[\lim_{x \to \infty} \frac{\sqrt{1 + x^4} + \left( 1 + x^2 \right)}{x^2}\]  is

\[\lim_{x \to 0} \frac{\sqrt{1 + x} - 1}{x}\] is equal to 

\[\lim_{x \to \pi/3} \frac{\sin \left( \frac{\pi}{3} - x \right)}{2 \cos x - 1}\] is equal to 

\[\lim_{x \to 3} \frac{\sum^n_{r = 1} x^r - \sum^n_{r = 1} 3^r}{x - 3}\]is real to

\[\lim_{n \to \infty} \frac{1 - 2 + 3 - 4 + 5 - 6 + . . . . + \left( 2n - 1 \right) - 2n}{\sqrt{n^2 + 1} + \sqrt{n^2 - 1}}\] is equal to 

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 

\[\lim_{n \to \infty} \frac{n!}{\left( n + 1 \right)! + n!}\]  is equal to

\[\lim_{x \to \pi/4} \frac{4\sqrt{2} - \left( \cos x + \sin x \right)^5}{1 - \sin 2x}\] is equal to 

\[\lim_{x \to 2} \frac{\sqrt{1 + \sqrt{2 + x} - \sqrt{3}}}{x - 2}\] is equal to 

\[\lim_{x \to \infty} a^x \sin \left( \frac{b}{a^x} \right), a, b > 1\] is equal to 

\[\lim_{x \to 0} \frac{8}{x^8}\left\{ 1 - \cos \frac{x^2}{2} - \cos \frac{x^2}{4} + \cos \frac{x^2}{2} \cos \frac{x^2}{4} \right\}\] is equal to 

If α is a repeated root of ax² + bx + c = 0, then \[\lim_{x \to \alpha} \frac{\tan \left( a x^2 + bx + c \right)}{\left( x - \alpha \right)^2}\]

The value of \[\lim_{x \to 0} \frac{\sqrt{a^2 - ax + x^2} - \sqrt{a^2 + ax + x^2}}{\sqrt{a + x} - \sqrt{a - x}}\] 

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 

\[\lim_\theta \to \pi/2 \frac{1 - \sin \theta}{\left( \pi/2 - \theta \right) \cos \theta}\] is equal to 

The value of \[\lim_{x \to \pi/2} \left( \sec x - \tan x \right)\]is 

The value of \[\lim_{x \to \infty} \frac{n!}{\left( n + 1 \right)! - n!}\] 

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 

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 

The value of \[\lim_{n \to \infty} \left\{ \frac{1 + 2 + 3 + . . . + n}{n + 2} - \frac{n}{2} \right\}\] 

\[\lim_{x \to 1} \left[ x - 1 \right]\] where [.] is the greatest integer function, is equal to 

\[\lim_{x \to \infty} \frac{\left| x \right|}{x}\]  is equal to 

\[\lim_{x \to 0} \frac{\left| \sin x \right|}{x}\]

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