91Ó°ÊÓ

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

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

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

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}

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

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

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

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