Chapter 2

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

$$ \lim _{x \rightarrow 0} \frac{10^{x}-1}{x^{2}}\\{\text { Ans. }+\infty,-\infty\\} $$

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

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

$$ \left.\lim _{x \rightarrow 0} \frac{\sin x-x+\frac{x^{3}}{6}}{x^{5}} \text { . Ans. } \frac{1}{120}\right\\} $$

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

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

$$ \lim _{x \rightarrow 0} \frac{e^{x}-e^{-x}+2 \sin x-4 x}{x^{5}} \text { \\{Ans. } \frac{1}{30} $$

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

$$ \left.\lim _{x \rightarrow 0} \frac{e^{x}-\frac{x^{3}}{6}-\frac{x^{2}}{2}-x-1}{\cos x+\frac{x^{2}}{5}-1} \text { \\{Ans. } 1\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{\ln (1+x)^{4}-4 x+2 x^{2}-\frac{4}{3} x^{3}+x^{4}}{6 \sin x-6 x+x^{3}} \text { \\{Ans. 16\\} } $$

$$ \left.\lim _{x \rightarrow 0} \frac{(1+2 x)^{7}-(1+3 x)^{5}}{x} \text { \\{Ans. }-1\right\\} $$

$$ \left.\lim _{x \rightarrow 0} \frac{(1+x)^{8}-(1+2 x)^{4}}{x^{2}} \text { \\{Ans. } 4\right\\} $$

$$ \left.\lim _{x \rightarrow 0} \frac{\sqrt[3]{1+x}-\sqrt[4]{1+2 x}}{x} \text { \\{Ans. }-\frac{1}{6}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sqrt{1+4 x}-\sqrt[3]{1+6 x}}{x^{2}}\\{\text { Ans. } 2 $$

$$ \left.\lim _{x \rightarrow a} \frac{\sqrt[3]{x}-\sqrt[3]{a}}{\sqrt{x}-\sqrt{a}} \text { \\{ Ans. } \frac{2}{3 \sqrt{a}}\right\\} $$

$$ \lim _{x \rightarrow 2} \frac{\sqrt{x+7}-3 \sqrt{2 x-3}}{\sqrt[3]{x+6}-2 \sqrt[3]{3 x-5}}\left\\{\text { Ans. } \frac{34}{23}\right\\} $$

$$ \left.\lim _{x \rightarrow 3} \log _{a} \frac{x-3}{\sqrt{x+6}-3} \text { \\{Ans. } \log _{a} 6\right\\} $$

$$ \left.\lim _{x \rightarrow \pm \infty} \sqrt{9 x^{2}+1}-3 x \text { \\{Ans. } 0,+\infty\right\\} $$

$$ \lim _{x \rightarrow+\infty} \frac{2 \sqrt{x}+3 \sqrt[3]{x}+5 \sqrt[5]{x}}{\sqrt{3 x-2}+\sqrt[3]{2 x-3}}\left\\{\text { Ans. } \frac{2}{\sqrt{3}}\right\\} $$

$$ \lim _{x \rightarrow-\infty} \sqrt{2 x^{2}-3}-5 x\\{\text { Ans. }+\infty\\} $$

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

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

$$ \lim _{x \rightarrow 0} \frac{\sqrt[k]{1+x}-1}{x}\left(k \text { is positive integer) }\left\\{\text { Ans. } \frac{1}{8}\right\\}\right. $$

$$ \lim _{x \rightarrow \frac{\pi}{5}} \frac{\sin \left(x-\frac{\pi}{6}\right)}{\sqrt{3}-2 \cos x}\\{\text { Ans. } 1\\} $$

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

$$ \lim _{x \rightarrow \infty} \frac{a x^{2}+b x+c}{A x^{2}+B x+C} \cdot\left\\{\text { Ans. } \frac{a}{A}\right\\} $$

$$ \left.\lim _{x \rightarrow a} \frac{\sqrt{a+2 x}-\sqrt{3 x}}{\sqrt{3 a+x}-2 \sqrt{x}} . \text { \\{Ans. } \frac{2}{3 \sqrt{3}}\right\\} $$

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

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

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

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

$$ \lim _{x \rightarrow 0} \frac{\ln (5+x)-\ln (5-x)}{x} .\left\\{\text { Ans. } \frac{2}{5}\right\\} $$

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

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

$$ \lim _{x \rightarrow x}\left(\frac{x^{2}+4 x-3}{x^{2}-2 x+5}\right)^{x} \cdot\left\\{\text { Ans. } e^{6}\right\\} $$

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

$$ \lim _{x \rightarrow \infty}\left(\frac{2 x^{2}+3}{2 x^{2}+5}\right)^{8 x^{2}+3} \quad\left\\{\text { Ans. } e^{-8}\right\\} $$

$$ \lim _{x \rightarrow 0}\left(\frac{1+5 x^{2}}{1+3 x^{2}}\right)^{\frac{1}{x^{2}}} \cdot\left\\{\text { Ans. } e^{2}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{x\left(5^{x}-1\right)}{1-\cos x} .\\{\text { Ans. } 2 \ln 5\\} $$

$$ \left.\lim _{x \rightarrow 0} \frac{(1-\cos 2 x) \sin 5 x}{x^{2} \sin 3 x} \text { . \\{Ans. } \frac{10}{3}\right\\} $$

$$ \lim _{x \rightarrow 0} \frac{\sin 4 x}{1-\sqrt{1-x}} \cdot\\{\text { Ans. } 8\\} $$

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

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

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

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

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

$$ \lim _{x \rightarrow 3} \frac{x-3}{|x-3|} \text { . }\\{\text { Ans. } 1,-1\\} $$

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

$$ \lim _{x \rightarrow 1}(1+\sin \pi x)^{\cot \pi x}\left\\{\text { Ans. } \frac{1}{c}\right\\} $$