91Ó°ÊÓ

Evaluate: \(\lim _{x \rightarrow 0}\left(\frac{\sin \\{x\\}}{\\{x\\}}\right)\), where \\{\\}\(=\) F.I.F

\(\lim _{x \rightarrow 0} \frac{64^{x}-32^{x}-16^{x}+4^{x}+2^{x}-1}{(\sqrt{(15+\cos x)-4}) \sin x}=\) (a) \(-96(\log 2)^{3}\) (b) \(48(\log 2)^{2}\) (c) \(\log 2\) (d) None

Evaluate: \(\lim _{n \rightarrow \infty} \prod_{r=3}^{n}\left(\frac{r^{3}-8}{r^{3}+8}\right)\)

Standard Result Method (SRM) Evaluate: \(\lim _{x \rightarrow 0}\left(\frac{\sqrt[3]{1+x^{2}}-\sqrt[4]{1-2 x}}{x+x^{2}}\right)\)

Standard Result Method (SRM) Evaluate: \(\lim _{x \rightarrow 1}\left(\frac{\sqrt[3]{7+x^{3}}-\sqrt[2]{3+x^{2}}}{x-1}\right)\)

Evaluate: \(\lim _{n \rightarrow \infty}\left\\{\cos \left(\frac{x}{2}\right) \cos \left(\frac{x}{4}\right) \cos \left(\frac{x}{8}\right) \ldots \cos \left(\frac{x}{2^{n}}\right)\right\\}\)

\(\lim _{x \rightarrow 0} \frac{8^{x}-4^{x}-2^{x}+1}{(\sqrt{2}-\sqrt{1+\cos x})}=\) (a) \((4 \sqrt{2}) \log 2\) (b) \((8 \sqrt{2}) \log 2\) (c) \((8 \sqrt{2})(\log 2)^{2}\) (d) None

Find the value of \(\lim _{x \rightarrow 0}\left\\{\frac{32}{x^{8}}\left(1-\cos \left(\frac{x^{2}}{2}\right)-\cos \left(\frac{x^{2}}{4}\right)+\cos \left(\frac{x^{2}}{2}\right) \cos \left(\frac{x^{2}}{4}\right)\right)\right\\}\)

\(\lim _{x \rightarrow 0}\left\\{\tan \left(\frac{\pi}{4}+x\right)\right\\}^{1 / x}=\) (a) \(e\) (b) \(e^{2}\) (c) \(\sqrt{e}\) (d) \(\frac{1}{\sqrt{e}}\)

Evaluate: \(\lim _{x \rightarrow \pi^{\prime}}\left(\frac{2^{\cot x}+3^{\operatorname{coc} x}-5^{1+\cot x}+10}{\left(4^{\cot x}\right)^{1 / 2}+\left(27^{\cot x}\right)^{1 / 3}-5^{\cot x}+20}\right)\)