Chapter 2

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

$$ \lim _{x \rightarrow \infty} \frac{1-2 x}{\sqrt[3]{1+8 x^{3}}}+2^{-x^{2}}\\{\text { Ans. }-1\\} $$

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

$$ \lim _{x \rightarrow-1} \frac{\sqrt[3]{1+2 x}+1}{\sqrt{2+x}+x}\left\\{\text { Ans, } \frac{4}{9}\right\\} $$

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

$$ \left.\lim _{x \rightarrow 0} \frac{\sqrt{x+4}-2}{\sin 5 x} \text { \\{Ans. } \frac{1}{20}\right\\} $$

$$ \lim _{x \rightarrow \infty}\left(\frac{x^{2}+5 x+3}{x^{2}+x+2}\right)^{x} \cdot\left\\{\text { Ans. } e^{4}\right\\} $$

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

$$ \lim _{\alpha \rightarrow \pi} \frac{\sin \alpha}{1-\frac{a^{2}}{x^{2}}}\left\\{\text { Ans. } \frac{\pi}{2}\right\\} $$

$$ \lim _{x \rightarrow \frac{\pi}{4}} \tan 2 x \cdot \tan \left(\frac{\pi}{4}-x\right)\left\\{\text { Ans. } \frac{1}{2}\right\\} $$

$$ \lim _{x \rightarrow \frac{\pi}{4}} \frac{\tan ^{3} x-3 \tan x}{\cos \left(x+\frac{\pi}{6}\right)}\\{\text { Ans. }-24\\} $$

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

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

$$ \left.\lim _{x \rightarrow \infty}\left(\frac{3 x^{2}+2 x+1}{x^{2}+x+2}\right)^{\text {fint }} \text { \\{Ans. } 9\right\\} $$

$$ \lim _{x \rightarrow \frac{1}{4}} \frac{\ln (\tan x)}{1-\cot x}\\{\text { Ans. } 1\\} $$

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

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

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

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

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

$$ \lim _{x \rightarrow \infty} \frac{(x+1)^{10}+(x+2)^{10}+\ldots+(x+100)^{10}}{x^{10}+10^{10}}\\{\text { Ans. } 100\\} $$

$$ \lim _{x \rightarrow+\infty} \frac{\sqrt{x^{2}+1}+\sqrt{x}}{\sqrt[4]{x^{3}+x}-x} \text { (Ans. -1\\} } $$

$$ \lim _{x \rightarrow \infty} \frac{\sqrt{x^{2}+1}-\sqrt[3]{x^{2}+1}}{\sqrt[4]{x^{4}+1}-\sqrt[5]{x^{4}+1}} \text { (Ans. 1\\} } $$

$$ \lim _{x \rightarrow+\infty} \frac{\sqrt[5]{x^{7}+3}+\sqrt[4]{2 x^{3}-1}}{\sqrt[6]{x^{8}+x^{7}+1-x}}\\{\text { Ans. } \infty\\} $$

$$ \lim _{x \rightarrow \infty} \frac{\sqrt[3]{x^{4}+3}-\sqrt[5]{x^{3}+4}}{\sqrt[3]{x^{7}+1}}\\{\text { Ans. } 0\\} $$

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

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

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

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

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

$$ \lim _{x \rightarrow \pm \infty} \sqrt{(x+a)(x+b)}-x\left\\{\text { Ans. } \frac{a+b}{2},+\infty\right\\} $$

$$ \left.\lim _{x \rightarrow 1} \sqrt{x^{2}-2 x-1}-\sqrt{x^{2}-7 x+3} \text { \\{Ans. } \frac{5}{2},-\frac{5}{2}\right\\} $$

$$ \lim _{x \rightarrow \infty} \sqrt[3]{(x+1)^{2}}-\sqrt[3]{(x-1)^{2}} \quad\\{\text { Ans. } 0\\} $$

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

$$ \lim _{\alpha \rightarrow 0} \frac{\tan \alpha}{\sqrt[3]{(1-\cos \alpha)^{2}}}\\{\text { Ans. }+\infty,-\infty\\} $$

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

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

$$ \lim _{\alpha \rightarrow 0} \frac{(1-\cos \alpha)^{2}}{\tan ^{3} \alpha-\sin ^{3} \alpha}\\{\text { Ans. }+\infty,-\infty\\} $$

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

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

$$ \lim _{x \rightarrow \frac{\pi}{2}} \frac{\cos x}{\sqrt[3]{(1-\sin x)^{2}}}\\{\text { Ans. }-\infty,+\infty\\} $$

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

$$ \lim _{y \rightarrow a} \sin \frac{y-a}{2} \cdot \tan \frac{\pi y}{2 a}\left\\{\text { Ans. }-\frac{a}{x}\right\\} $$

$$ \lim _{x \rightarrow \frac{4}{7}} \frac{\cos x-\sin x}{\cos 2 x}\left\\{\text { Ans. } \frac{1}{\sqrt{2}}\right\\} $$

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

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

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

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

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

$$ \lim _{\alpha \rightarrow \beta} \frac{\sin ^{2} \alpha-\sin ^{2} \beta}{\alpha^{2}-\beta^{2}}\left\\{\text { Ans. } \frac{\sin 2 \beta}{2 \beta}\right\\} $$