Chapter 2

$$ \begin{aligned} &\text { If }\\\ &f(x)=x \sin \frac{1}{x}, \quad x \neq 0\\\ &=1, \quad x=0\\\ &\text { then find } \left.\lim _{x \rightarrow 0} f(x) \text { . \\{Ans. } 0\right\\} \end{aligned} $$

$$ \lim _{x \rightarrow 2} \frac{\sqrt{1+\sqrt{2+x}}-\sqrt{3}}{x-2} .\left\\{\text { Ans. } \frac{1}{8 \sqrt{3}}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt{1-\cos x^{2}}}{1-\cos x} .\\{\text { Ans. } \sqrt{2}\\} $$

$$ \lim _{x \rightarrow 0} \frac{\tan x-x}{x^{3}} . \text { \\{Ans. } \frac{1}{3} $$

$$ \lim _{x \rightarrow \infty} \sqrt{x+\sqrt{x+\sqrt{x}}}-\sqrt{x} .\left\\{\text { Ans. } \frac{1}{2}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{3 x+|x|}{7 x-5|x|} \cdot\left\\{\text { Ans. } 2, \frac{1}{6}\right\\} $$

$$ \left.\lim _{x \rightarrow-\infty} \frac{\left(3 x^{4}+2 x^{2}\right) \sin \frac{1}{x}+|x|^{3}+5}{|x|^{3}+|x|^{2}+|x|+1} \text { . Ans. }-2\right\\} $$

$$ \left.\lim _{x \rightarrow a} \frac{a \sin x-x \sin a}{x^{2}-a x}(a \neq 0) \text { . \\{ Ans. } \cos a-\frac{\sin a}{a}\right\\} $$

$$ \lim _{x \rightarrow \infty} \frac{\ln x}{x}\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow 0} x \ln x\\{\text { Ans. } 0\\} $$

$$ \left.\lim _{x \rightarrow 0} x^{x} \text { \\{Ans. } 1\right\\} $$

$$ \left.\lim _{x \rightarrow 0} x^{x} \text { \\{Ans. } 1\right\\} $$

Show that the following limits cannot be evaluated by L' Hospital's rule:- i. \(\lim _{x \rightarrow 0} \frac{x^{2} \sin \frac{1}{x}}{\sin x}\\{\) Ans. 0\(\\}\) ii. \(\lim _{x \rightarrow \infty} \frac{2+2 x+\sin 2 x}{(2 x+\sin 2 x) e^{\sin x}}\) \\{Ans. no limit\\} iii. \(\lim _{x \rightarrow \infty} \frac{x-\sin x}{x+\sin x}\\{\) Ans. 1\(\\}\) iv. \(\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan x}{\sec x}\\{\) Ans. 1\(\\}\)

\lim _{x \rightarrow 0} \frac{\ln \left(1+x^{2}\right)}{\cos 3 x-e^{-x}}\\{\text { Ans. } 0\\}

\lim _{x \rightarrow 0} \frac{\sin 3 x^{2}}{\ln \cos \left(2 x^{2}-x\right)}\\{\text { Ans. }-6\\}

\lim _{x \rightarrow+\infty} \frac{\log _{a} x}{x^{k}}(k>0)\\{\text { Ans. } 0\\}

\lim _{x \rightarrow 1} \frac{1}{\ln x}-\frac{1}{x-1}\left\\{\text { Ans. } \frac{1}{2}\right\\}

$$ \lim _{x \rightarrow 0} \cot x-\frac{1}{x}\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow 0} \frac{1}{x}-\frac{1}{e^{x}-1}\left\\{\text { Ans. } \frac{1}{2}\right\\} $$

$$ \lim _{x \rightarrow 0} x^{n} \ln x(n>0)\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow 0}\left(\frac{1}{x}\right)^{\sin x} \quad\\{\text { Ans. } 1\\} $$

$$ \lim _{x \rightarrow 0} x^{\frac{1}{\ln \left(e^{x}-1\right)}}\\{\text { Ans. } e\\} $$

$$ \lim _{x \rightarrow \frac{\pi}{2}}(\tan x)^{\cot x} \quad\\{\text { Ans. } 1\\} $$

$$ \left.\lim _{x \rightarrow 2} \frac{\ln \left(x^{2}-3\right)}{x^{2}+3 x-10} \text { \\{Ans. } \frac{4}{7}\right\\} $$

$$ \left.\lim _{x \rightarrow 1} \frac{a^{\ln x}-x}{\ln x} \text { \\{Ans. } \ln a-1\right\\} $$

$$ \lim _{x \rightarrow 1} \frac{1-4 \sin ^{2}\left(\frac{\pi x}{6}\right)}{1-x^{2}}\left\\{\text { Ans. } \frac{\pi \sqrt{3}}{6}\right\\} $$

$$ \lim _{x \rightarrow a} \sin ^{-1}\left(\frac{x-a}{a}\right) \cot (x-a)\left\\{\text { Ans. } \frac{1}{a}\right\\} $$

$$ \lim _{x \rightarrow+\infty}\left(\pi-2 \tan ^{-1} x\right) \ln x\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow+\infty}\left(\pi-2 \tan ^{-1} x\right) \ln x\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow 0} \frac{1}{x^{2}}-\cot ^{2} x\left\\{\text { Ans. } \frac{2}{3}\right\\} $$

$$ \lim _{x \rightarrow \infty} x-x^{2} \ln \left(1+\frac{1}{x}\right)\left\\{\text { Ans. } \frac{1}{2}\right\\} $$

$$ \lim _{x \rightarrow \infty} x^{2}\left[\cosh \frac{a}{x}-1\right]\left\\{\text { Ans. } \frac{a^{2}}{2}\right\\} $$

$$ \lim _{x \rightarrow 0}\left(\frac{5}{2+\sqrt{9+x}}\right)^{\frac{1}{\sin x}}\left\\{\text { Ans. } e^{-\frac{1}{30}}\right\\} $$

$$ \lim _{x \rightarrow 0}(\ln \cot x)^{\tan x}\\{\text { Ans. } 1\\} $$

$$ \lim _{x \rightarrow 0} \frac{1-\sqrt{1+x^{2}} \cos x}{\tan ^{4} x}\left\\{\text { Ans. } \frac{1}{3}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt[3]{1+3 x}-\sqrt{1+2 x}}{x^{2}}\left\\{\text { Ans. }-\frac{1}{2}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{e^{x} \sin x-x(1+x)}{x^{3}}\left\\{\text { Ans. } \frac{1}{3}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{e^{\alpha x}-\cos \alpha x}{e^{\beta x}-\cos \beta x}\left\\{\text { Ans. } \frac{\alpha}{\beta}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{x-\tan ^{-1} x}{x^{3}}\left\\{\text { Ans. } \frac{1}{3}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{e^{a \sqrt{x}}-1}{\sqrt{\sin b x}}\left\\{\text { Ans. } \frac{a}{\sqrt{b}}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{x-\sin x}{x-\tan x}\left\\{\text { Ans. }-\frac{1}{2}\right\\} $$

$$ \lim _{x \rightarrow \infty} \frac{\pi-2 \tan ^{-1} x}{\ln \left(1+\frac{1}{x}\right)}\\{\text { Ans. } 2\\} $$

$$ \lim _{x \rightarrow 0} \frac{a^{x}-b^{x}}{c^{x}-d^{x}}\left\\{\text { Ans. } \frac{\ln \frac{a}{b}}{\ln \frac{c}{d}}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{a^{x}-b^{x}}{x \sqrt{1-x^{2}}}\left\\{\text { Ans. } \ln \frac{a}{b}\right\\} $$

$$ \lim _{x \rightarrow a} \frac{\cos x \ln (x-a)}{\ln \left(e^{x}-e^{a}\right)}\\{\text { Ans. } \cos a\\} $$

$$ \lim _{x \rightarrow 0} \frac{e^{\tan x}-e^{x}}{\tan x-x}\\{\text { Ans. } 1\\} $$

$$ \lim _{x \rightarrow 0} \frac{e^{x^{3}}-1-x^{3}}{\sin ^{6} 2 x}\left\\{\text { Ans. } \frac{1}{128}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt[3]{1+\tan ^{-1} 3 x}-\sqrt[3]{1-\sin ^{-1} 3 x}}{\sqrt{1-\sin ^{-1} 2 x}-\sqrt{1+\tan ^{-1} 2 x}} \text { \\{Ans. -1\\} } $$

$$ \lim _{x \rightarrow 0} \frac{2 x^{2}-2 e^{x^{2}}+2 \cos x^{\frac{3}{2}}+\sin ^{3} x}{x^{4}}\\{\text { Ans. }-1\\} $$

$$ \lim _{x \rightarrow 0} \frac{\ln \sin 2 x}{\ln \sin x}\\{\text { Ans. } 1\\} $$