Chapter 2

$$ \lim _{x \rightarrow 0} \frac{\log _{\sec \frac{x}{2}} \cos x}{\log _{\sec x} \cos \frac{x}{2}}\\{\text { Ans. } 16\\} $$

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

$$ \lim _{x \rightarrow \infty}\left(1+\frac{1}{a+b x}\right)^{c+d x}\left(a, b, c, d \text { are positive) }\left\\{\text { Ans. } e^{\frac{d}{b}}\right\\}\right. $$

$$ \lim _{n \rightarrow \infty} \frac{n^{K} \cos n !}{n+1}(0

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

$$ \text { Find } \lim _{m \rightarrow \infty}\left(\cos \frac{x}{m}\right)^{m} \text { without using } \mathrm{L} \text { ' Hospital's rule. \\{Ans. 1\\} } $$

$$ \text { Find } \lim _{x \rightarrow \infty} \frac{x^{2} \sin \left(\frac{1}{x}\right)-x}{1-|x|} \cdot\\{\text { Ans. } 0\\} $$

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

If \(\begin{aligned} f(x) &=x^{4}, & & x^{2}<1 \\ &=x, & & x^{2} \geq 1 \end{aligned}\) Discuss the existence of limit at \(x=1\) and \(x=-1\).

$$ \left.\lim _{x \rightarrow 0}\left[\frac{\cos x+\cos 2 x+\ldots+\cos n x}{n}\right]^{\frac{1}{x^{2}}} \text { \\{Ans. } e^{-\frac{(n+1)(2 n+1)}{12}}\right\\} $$

$$ \text { If } \sin x

$$ \text { If } \left.x^{2} \leq f(x) \leq|x| \text { in the neighbourhood of } 0, \text { find } \lim _{x \rightarrow 0} f(x) \text { . \\{Ans. } 0\right\\} $$

$$ \text { If }|f(x)| \leq|x| \forall x, \text { find } \lim _{x \rightarrow 0} f(x) .\\{\text { Ans. } 0\\} $$

$$ \text { If } \left.f(x)>\ln x \forall x>0, \text { find } \lim _{x \rightarrow \infty} f(x) \text { . Ans. } \infty\right\\} $$

$$ \lim _{x \rightarrow \pm \infty} \frac{a^{x}}{a^{x}+1}(a>0) $$

$$ \lim _{x \rightarrow \pm \infty} \frac{a^{x}-a^{-x}}{a^{x}+a^{-x}}(a>0) $$

$$ \lim _{x \rightarrow \infty} \sqrt{a^{2} x^{2}+a x+1}-\sqrt{a^{2} x^{2}+1} $$

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

$$ \lim _{h \rightarrow 0} \frac{\sqrt{x+h}-\sqrt{x}}{h} $$

$$ \lim _{x \rightarrow 0} \frac{\sin \left(x^{n}\right)}{(\sin x)^{m}}(m \text { and } n \text { are positive integers) } $$

How do the roots of the equation \(a x^{2}+b x+c=0\) change when \(b\) and \(c\) retain constant values \((b \neq 0)\) and \(a\) tends to zero. \\{ns. One root tends to \(-\frac{c}{b}\) and the other root tends to \(\left.\pm \infty\right\\}\)

$$ \lim _{x \rightarrow+\infty} \frac{x^{b}}{a^{x}} $$

$$ \lim _{x \rightarrow \infty} \frac{(a x+1)^{n}}{x^{n}+A} \text { . Consider separately the cases when } n \text { is (1) positive integer, (2) negative integer, (3) zero } $$

$$ \text { Find the constants } \left.a \text { and } b \text { such that } \lim _{x \rightarrow \infty} \frac{x^{2}+1}{x+1}-a x-b=0 \quad \text { Ans. } a=1, b=-1\right\\} $$

$$ \text { If } \lim _{x \rightarrow \infty} \frac{x^{2}+1}{x+1}-a x-b=\infty, \text { find the value of } a \text { and } b \text { . } $$

$$ \text { If } \lim _{x \rightarrow \infty} \frac{x^{2}-1}{x+1}-a x-b=2, \text { find the values of } a \text { and } b \text { . } $$

$$ \text { If } \lim _{x \rightarrow \infty} \frac{x^{3}+1}{x^{2}+1}-a x-b=2 \text { , then find } a \text { and } b \text { . } $$

$$ \text { Find the constants } \left.a \text { and } b \text { such that } \lim _{x \rightarrow-\infty} \sqrt{x^{2}-x+1}-a x-b=0 \text { \\{ Ans. } a=-1, b=\frac{1}{2}\right\\} $$

$$ \text { If } \left.\lim _{x \rightarrow 0} \frac{a e^{x}+b e^{-x}+c x}{x^{3}}=1 \text { , find the values of } a, b, c \text { . \\{Ans. } a=3, b=-3, c=-6\right\\} $$

$$ \text { If } \lim _{x \rightarrow 0} \frac{\sin x+a e^{x}+b e^{-x}+c \ln (1+x)}{x^{3}} \text { is finite, find } a, b, c \text { and the value of the limit. } $$

$$ \text { If } \lim _{x \rightarrow \infty} \frac{e^{-x}+x \ln x+x^{2} \sin a x}{1+x^{2}} \text { is finite, find } a \text { and the value of the limit. } $$

$$ \text { If } \lim _{x \rightarrow 0} \frac{x^{2}+\ln (1-a x)+b \sin x}{x^{3}} \text { is finite, find } a, b \text { and the value of the limit. } $$

$$ \text { If } \lim _{x \rightarrow 0} \frac{\sinh 3 x+a \sinh 2 x+b \sinh x}{x^{5}} \text { is finite, find } a, b \text { and the value of the limit. } $$

Given \(f(x)=\frac{\sin x}{x}, \quad x<0\) \(=a x+b, \quad x>0 .\) If \(\lim _{x \rightarrow 0} f(x)\) exists, find \(a, b\) and the value of the limit.

\text { If } \lim _{x \rightarrow 0} \frac{\\{(a-n) n x-\tan x\\} \sin x}{x^{2}}=0, \text { where } n \text { is non-zero real no., then find the value of } a

\begin{aligned} &\text { Sketch the graph of the function }\\\ &f(x)=\lim _{n \rightarrow \infty} \sqrt[n]{1+x^{n}}, \quad x \geq 0 \end{aligned}

$$ \text { Sketch the graph of the function } f(x)=\lim _{n \rightarrow \infty} \sin ^{2 n} x $$

$$ \text { Determine the function } f(x)=\lim _{n \rightarrow \infty}\left(\cos \frac{x}{2} \cdot \cos \frac{x}{4} \cdots \cdots \cdots \cos \frac{x}{2^{n}}\right) \text { . } $$

$$ \text { Determine the function } f(x)=\lim _{n \rightarrow \infty} \lim _{m \rightarrow \infty}(\cos m ! \pi x)^{n} $$

$$ \text { Find the value of } k \text { if } \lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}=\lim _{x \rightarrow k} \frac{x^{3}-k^{3}}{x^{2}-k^{2}} \text { . } $$

$$ \text { If } \lim _{x \rightarrow-a} \frac{x^{9}+a^{9}}{x+a}=9, \text { find the value of } a \text { . } $$

$$ \text { Let } f(x)=\frac{1}{\sqrt{18-x^{2}}} \text { . Find the value of } \lim _{x \rightarrow 3} \frac{f(x)-f(3)}{x-3} \text { . } $$

$$ \text { Find the polynomial function } f(x) \text { of least degree satisfying } \lim _{x \rightarrow 0}\left(1+\frac{f(x)}{x^{3}}\right)^{\frac{1}{x}}=e^{2} \text { . } $$

$$ \text { Find a polynomial } f(x) \text { of least degree such that } \lim _{x \rightarrow 0}\left(2+\frac{f(x)}{x^{2}}\right)^{\frac{1}{x}}=e^{2} \text { . } $$

$$ \text { Evaluate } \lim _{x \rightarrow 0} \frac{x-\sin x}{x^{3}} \text { without using series and } L \text { ' Hospital's rule. } $$

$$ \text { The function } f(x) \text { satisfies the condition } f(x)=\frac{1}{3}\left[f(x+1)+\frac{5}{f(x+2)}\right] \text { . Find } \lim _{x \rightarrow \infty} f(x) \text { . } $$

$$ \text { Find } \lim _{n \rightarrow \infty} n^{2} \sqrt{\left(1-\cos \frac{1}{n}\right) \sqrt{\left(1-\cos \frac{1}{n}\right) \sqrt{\left(1-\cos \frac{1}{n}\right) \cdots \cdots \infty}}} $$

$$ \text { Find } \lim _{x \rightarrow 0} \frac{\cos ^{2}\left(1-\cos ^{2}\left(1-\cos ^{2}\left(1-\cos ^{2} \cdots \cdots-\cos ^{2}\left(1-\cos ^{2} x\right)\right)\right)\right)}{\sin \left(\pi \frac{\sqrt{x+4}-2}{x}\right)} \text { . } $$

$$ \lim _{n \rightarrow \infty} \frac{1-2+3-4+\cdots \cdots \cdots+(2 n-1)-2 n}{\sqrt{n^{2}+1}} $$

$$ \lim _{n \rightarrow \infty}(1+x)\left(1+x^{2}\right)\left(1+x^{4}\right) \cdots \cdots \cdots \cdot\left(1+x^{2^{n}}\right), \text { where }|x|<1 $$