Chapter 2

$$ \lim _{x \rightarrow \infty} \frac{\ln ^{2} x-\ln x+1}{\ln x+3}\\{\text { Ans. } \infty\\} $$

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

$$ \lim _{x \rightarrow 1} \frac{\sqrt[3]{x^{2}}-2 \sqrt[3]{x}+1}{(x-1)^{2}}\left\\{\text { Ans. } \frac{1}{9}\right. $$

$$ \left.\lim _{x \rightarrow 1} \frac{\sqrt[3]{x}-1}{\sqrt[4]{x}-1} \text { \\{Ans. } \frac{4}{3}\right\\} $$

$$ \lim _{x \rightarrow-1} \frac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}\left\\{\text { Ans. } \frac{5}{3}\right. $$

$$ \lim _{x \rightarrow-1} \frac{x+1}{\sqrt[4]{x+17}-2}\\{\text { Ans. } 32\\} $$

$$ \lim _{x \rightarrow \frac{\pi}{6}} \frac{2 \sin ^{2} x+\sin x-1}{2 \sin ^{2} x-3 \sin x+1}\\{\text { Ans. }-3\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sin 3 x}{x}\\{\text { Ans. } 3\\} $$

$$ \text { 115. } \lim _{x \rightarrow 0} \frac{\tan k x}{x}\\{\text { Ans. } k\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sin \alpha x}{\sin \beta x}\left\\{\text { Ans. } \frac{a}{\beta}\right\\} $$

$$ \left.\lim _{x \rightarrow 0} \frac{\tan 2 x}{\sin 5 x} \text { \\{Ans. } \frac{2}{5}\right\\} $$

$$ \lim _{x \rightarrow 1} \frac{\sin (\ln x)}{\ln x}\\{\text { Ans. } 1\\} $$

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

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

$$ \lim _{x \rightarrow 0} \frac{\sin ^{2} 2 x}{x^{2}}\\{\text { Ans. } 4\\} $$

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

$$ \lim _{x \rightarrow 1} \frac{\cos \left(\frac{\pi x}{2}\right)}{1-x}\left\\{\text { Ans. } \frac{\pi}{2}\right\\} $$

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

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

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

$$ \lim _{x \rightarrow 0} \frac{x}{\tan ^{-1} 2 x} \cdot\left\\{\text { Ans, } \frac{1}{2}\right\\} $$

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

$$ \lim _{x \rightarrow-3} \frac{x^{2}-9}{\tan ^{-1}(x+3)}\\{\text { Ans. }-6\\} $$

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

$$ \lim _{x \rightarrow 1} \frac{x^{10}-1}{x-1}\\{\text { Ans. } 10\\} $$

$$ \left.\lim _{x \rightarrow-1} \frac{x^{13}+1}{x^{17}+1} \text { \\{Ans. } \frac{13}{17}\right\\} $$

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

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

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

$$ \lim _{x \rightarrow 0}\left(1+x^{2}\right)^{\frac{1}{x}}\\{\text { Ans. } 1\\} $$

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

$$ \left.\lim _{x \rightarrow \infty}\left(1+\frac{1}{x}\right)^{7 x} \text { \\{Ans, } e^{7}\right\\} $$

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

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

$$ \lim _{x \rightarrow \infty}\left(\frac{x+3}{x-1}\right)^{x+3} \cdot\left\\{\text { Ans. } e^{4}\right\\} $$

$$ \lim _{x \rightarrow 0}(1+\sin x)^{\operatorname{cosecx}}\\{\text { Ans. } e\\} $$

$$ \left.\lim _{x \rightarrow 0}\left(1+3 \tan ^{2} x\right)^{\cot ^{2} x} \text { \\{Ans. } e^{3}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{\ln (1+k x)}{x}\\{\text { Ans. } k\\} $$

$$ \left.\lim _{x \rightarrow 0} \frac{1}{\sin x} \ln (1+a \sin x) \text { \\{Ans. } a\right\\} $$

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

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

$$ \text { 150. } \lim _{x \rightarrow 0} \frac{e^{2 x}-1}{3 x}\left\\{\text { Ans. } \frac{2}{3}\right\\} $$

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

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

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

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

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

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

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

$$ \lim _{x \rightarrow 0} \frac{e^{4 x}-1}{\tan x}\\{\text { Ans. } 4\\} $$