Chapter 2

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

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

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

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

$$ \lim _{x \rightarrow 0} \frac{\sqrt{1+x \sin x}-\sqrt{\cos 2 x}}{\tan ^{2} \frac{x}{2}}\\{\text { Ans. } 6\\} $$

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

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

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

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

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

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

$$ \lim _{x \rightarrow \infty}\left(\frac{x+1}{x-2}\right)^{2 x-1}\left\\{\text { Ans. } e^{6}\right\\} $$

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

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

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

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

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

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

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

$$ \lim _{x \rightarrow \infty} x(\ln (x+a)-\ln x)\\{\text { Ans. } a\\} $$

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

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

$$ \lim _{x \rightarrow 0} \frac{\cos x-e^{-\frac{x^{2}}{2}}}{x^{4}}\left\\{\text { Ans. }-\frac{1}{12}\right\\} $$

$$ \lim _{x \rightarrow \pm \infty} \cosh x-\sinh x\\{\text { Ans. } 0,+\infty\\} $$

$$ \lim _{x \rightarrow \pm \infty} \tanh x \quad\\{\text { Ans. } 1,-1\\} $$

$$ \lim _{x \rightarrow \pm \infty} x\left(\sqrt{x^{2}+\sqrt{x^{4}+1}}-x \sqrt{2}\right)\\{\text { Ans. } 0,-\infty\\} $$

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

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

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

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

$$ \lim _{x \rightarrow \frac{\pi}{2}} \tan ^{2} x \cdot\left(\sqrt{2 \sin ^{2} x+3 \sin x+4}-\sqrt{\sin ^{2} x+6 \sin x+2}\right)\left\\{\text { Ans. } \frac{1}{12}\right\\} $$

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

$$ \lim _{x \rightarrow 0} \frac{1-\cos (1-\cos x)}{x^{4}}\left\\{\text { Ans. } \frac{1}{8}\right\\} $$

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

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

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

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

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

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

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

$$ \lim _{x \rightarrow \infty} \cos \sqrt{x+1}-\cos \sqrt{x} \quad\\{\text { Ans. } 0\\} $$

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

$$ \lim _{x \rightarrow \infty} \frac{3^{x}+4^{x}}{5^{x}+6^{x}}\\{\text { Ans. } 0\\} $$

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

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

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

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

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

$$ \lim _{x \rightarrow 1} 3+\frac{1}{1+7^{\frac{1}{1-x}}}\\{\text { Ans. } 4,3\\} $$

$$ \lim _{x \rightarrow 0} \frac{|x|}{x^{2}}\\{\text { Ans. } \infty\\} $$