Chapter 2

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

$$ \lim _{x \rightarrow+\infty} x^{n} e^{-x}(n>0)\\{\text { Ans. } 0\\} $$

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

$$ \left.\lim _{\varphi \rightarrow a}\left(a^{2}-\varphi^{2}\right) \tan \frac{\pi \varphi}{2 a} \text { \\{Ans. } \frac{4 a^{2}}{\pi}\right\\} $$

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

$$ \lim _{x \rightarrow \infty} \sqrt[3]{(a+x)(b+x)(c+x)}-x \quad\left\\{\text { Ans. } \frac{a+b+c}{3}\right\\} $$

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

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

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

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

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

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

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

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

$$ \left.\lim _{x \rightarrow 1} \frac{\sqrt[3]{7+x^{3}}-\sqrt{3+x^{2}}}{x-1} \text { (Ans. }-\frac{1}{4}\right\\} $$

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

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

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

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

$$ \left.\lim _{x \rightarrow 0}\left[\tan \left(\frac{\pi}{4}+x\right)\right]^{\frac{1}{x}} \text { \\{ Ans. } e^{2}\right\\} $$

$$ \left.\lim _{x \rightarrow \infty}\left(\frac{a^{\frac{1}{x}}+b^{\frac{1}{x}}}{2}\right)^{x} \quad(a, b>0) \text { \\{Ans. } \sqrt{a b}\right\\} $$

$$ \lim _{x \rightarrow 0}\left(\frac{a^{x}+b^{x}+c^{x}}{3}\right)^{\frac{1}{x}} \cdot\\{\text { Ans. } \sqrt[3]{a b c}\\} $$

$$ \lim _{x \rightarrow a}\left(\frac{\sin x}{\sin a}\right)^{\frac{1}{x-a}} \quad\left(a \neq k \pi, k \text { is an integer) }\left\\{\text { Ans. } e^{\cot a}\right\\}\right. $$

$$ \lim _{x \rightarrow 0} \frac{(1+x)^{\frac{1}{x}}+\frac{e x}{2}-e}{x^{2}}\left\\{\text { Ans. } \frac{11 e}{24}\right\\} $$

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

$$ \lim _{x \rightarrow \infty}\left(\frac{a_{1}^{\frac{1}{x}}+a_{2}^{\frac{1}{x}}+\ldots \ldots \ldots .+a_{n}^{\frac{1}{x}}}{n}\right)^{n x} $$

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

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

$$ \lim _{x \rightarrow 0} \frac{\cos \left(\frac{\pi}{2} \cos x\right)}{x^{2}} \text { \\{Ans. } \frac{\pi}{4} \text { . } $$

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

$$ \lim _{x \rightarrow 2} \frac{2^{x}+2^{3-x}-6}{\sqrt{2^{-x}}-2^{1-x}}\\{\text { Ans. } 8\\} $$

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

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

$$ \begin{aligned} &\lim _{x \rightarrow 2} \frac{(\sin \theta)^{x}+(\cos \theta)^{x}-1}{x-2} \\ &\text { \\{Ans. } \left.\sin ^{2} \theta \ln (\sin \theta)+\cos ^{2} \theta \ln (\cos \theta)\right\\} \end{aligned} $$

$$ \lim _{x \rightarrow \infty} x^{2}\left(1+\frac{1}{x}\right)^{x}-e x^{3} \ln \left(1+\frac{1}{x}\right) \text { \\{Ans. } \frac{e}{8} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt{a^{2}+a x+x^{2}}-\sqrt{a^{2}+a x}}{\ln \left(\cos \frac{x}{a}\right)}\\{\text { Ans. }-|a|\\} $$

$$ \lim _{x \rightarrow 0} \frac{x^{2}-\sin ^{2} x-\frac{x}{3}}{x^{6}}\left\\{\text { Ans. }-\frac{2}{45}\right\\} $$

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

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

$$ \left.\lim _{x \rightarrow a} \sqrt{a^{2}-x^{2}} \cot \left(\frac{\pi}{2} \sqrt{\frac{a-x}{a+x}}\right) \text { \\{Ans. } \frac{4 a}{\pi}\right\\} $$

$$ \left.\lim _{x \rightarrow 0} \frac{\left(a^{x}-1\right)}{x^{n} \sin x}\left(\frac{b \sin x-\sin b x}{\cos x-\cos b x}\right)^{n} \text { \\{ Ans. }\left(\frac{b}{3}\right)^{n} \ln a\right\\} $$

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

$$ \lim _{h \rightarrow 0} \frac{\sin (a+3 h)-3 \sin (a+2 h)+3 \sin (a+h)-\sin a}{h^{3}}\\{\text { Ans. }-\cos a\\} $$

$$ \lim _{x \rightarrow 0} \frac{x \sin (\sin x)-\sin ^{2} x}{x^{6}} \text { \\{Ans. } \frac{1}{18} $$

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

$$ \lim _{n \rightarrow \infty} \sin \pi \sqrt{n^{2}+1}(n \text { is an integer) }\\{\text { Ans. } 0\\} $$

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

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

$$ \lim _{x \rightarrow 0}[\ln (\sec x)]^{x} \quad\\{\text { Ans. } 1\\} $$

$$ \lim _{n \rightarrow \infty} \tan ^{n}\left[\frac{\pi-4}{4}+\left(1+\frac{1}{n}\right)^{\alpha}\right]\left\\{\text { Ans. } e^{2 \alpha}\right\\} $$