Chapter 2

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

$$ \lim _{x \rightarrow \infty} \frac{\tanh x}{x^{3}}\\{\text { Ans. } 0\\} $$

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

$$ \lim _{x \rightarrow 0} \operatorname{sgn}\left(x^{3}\right)\\{\text { Ans. } 1,-1\\} $$

$$ \lim _{x \rightarrow 0}(\operatorname{sgn} x)^{2} \quad\\{\text { Ans. } 1\\} $$

$$ \left.\lim _{x \rightarrow 0}(\operatorname{sgn} x)^{\operatorname{sgn} x} \text { . Ans. } 1,-1\right\\} $$

$$ \lim _{x \rightarrow 0} e^{\operatorname{sgn} x}\left\\{\text { Ans. } e, \frac{1}{e}\right\\} $$

$$ \lim _{x \rightarrow 0} e^{|\operatorname{sgn} x|}\\{\text { Ans. } e\\} $$

$$ \lim _{x \rightarrow 0}(\operatorname{sgn} x)^{e^{x}} \quad\\{\text { Ans. } 1\\} $$

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

$$ \left.\lim _{x \rightarrow 1} 3 x^{2}+4 x+5 . \text { \\{Ans. } 12\right\\} $$

$$ \lim _{x \rightarrow 1} \frac{4 x^{5}+9 x+7}{3 x^{6}+x^{3}+1}\\{\text { Ans. } 4\\} $$

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

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

$$ \lim _{x \rightarrow e} x^{2} \ln x \quad\left\\{\text { Ans. } e^{2}\right\\} $$

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

$$ \lim _{x \rightarrow 1} \ln (\sin \pi x-\cos \pi x) \quad\\{\text { Ans. } 0\\} $$

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

$$ \lim _{x \rightarrow 0} \operatorname{sgn}\\{\ln (\cos x)\\}\\{\text { Ans. }-1\\} $$

$$ \lim _{x \rightarrow 0} \operatorname{sgn}\\{\ln (1+x)\\}\\{\text { Ans. } 1,-1\\} $$

$$ \lim _{x \rightarrow \infty} x \ln (\operatorname{sgn} x)\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow 0}\\{\ln (\operatorname{sgn} x)\\}^{x} \quad\\{\text { Ans. } 0\\} $$

$$ \lim _{x \rightarrow \infty} x^{\ln (\operatorname{sgn} x)}\\{\text { Ans. } 1\\} $$

$$ \lim _{x \rightarrow \infty}(\operatorname{sgn} x)^{x} \quad\\{\text { Ans. } 1\\} $$

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

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

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

$$ \lim _{x \rightarrow 0} \ln x+e^{x}\\{\text { Ans. }-\infty\\} $$

$$ \lim _{x \rightarrow \frac{\pi}{2}} \tan x+\tan 2 x \quad\\{\text { Ans. }-\infty,+\infty\\} $$

$$ \lim _{x \rightarrow 0} \cos ^{-1} x \cdot \cot x+\tan ^{-1} x \quad\\{\text { Ans. }+\infty,-\infty\\} $$

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

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

$$ \lim _{x \rightarrow 1} \frac{\sqrt{x+3}}{\sqrt[3]{x+7}}\\{\text { Ans. } 1\\} $$

$$ \lim _{x \rightarrow \infty} x+e^{x} \quad\\{\text { Ans. } \infty\\} $$

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

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

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

$$ \lim _{x \rightarrow \infty} x^{x} \quad\\{\text { Ans. } \infty\\} $$

$$ \lim _{x \rightarrow 0}(\sin x)^{\cot x} \quad\\{\text { Ans. } 0\\} $$

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

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

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

$$ \lim _{x \rightarrow \frac{\pi}{2}} \frac{1}{1+e^{\tan x}}\\{\text { Ans. } 1,0\\} $$

$$ \lim _{x \rightarrow 0} \frac{\ln |x|}{\sin x}\\{\text { Ans. }-\infty, \infty\\} $$

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

$$ \begin{aligned} &\text { Given }\\\ &\begin{aligned} f(x) &=-2 x+3, & & x \leq 1 \\ &=3 x-5, & & x>1 \end{aligned} \end{aligned} $$

Given \(f(x)=x+2, \quad x<1\) \(\quad=4 x-1, \quad 1 \leq x<3\) \(\quad=x^{2}+5, \quad x>3\) Find:- i. \(\lim _{x \rightarrow 1} f(x) .\) Ans. 3\(\\}\) ii.

A function is defined as \(f(x)=1, \quad x \neq 0\) \(=2, \quad x=0\) Does the limit \(\lim f(x)\) exists? \\{Ans. Yes \(\\}\)

\text { Draw the graph of function } f(x)=\frac{|x|}{x} . \text { Is } f(0) \text { defined? Does } \lim _{x \rightarrow 0} f(x) \text { exist? \\{Ans. No, No\\} }

\begin{aligned} &\text { Given }\\\ &\begin{aligned} f(x) &=x, \quad x<0 \\ &=1, \quad x=0 \\ &=x^{2}, \quad x>0 \\ \text { Does } & \lim _{x \rightarrow 0} f(x) \text { exist? \\{ns. Yes\\} } \end{aligned} \end{aligned}