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Question Number 80653 by jagoll last updated on 05/Feb/20

$$\underset{x\to 0}{\lim }{\left(\frac{\sin x}{x}\right)}^{\frac{3}{x^2}}$$

Commented by john santu last updated on 05/Feb/20

$$\underset{x\to 0}{\lim }{(1+(\frac{\sin x}{x}-1))}^{\frac{3}{x^2}}=$$$$\underset{x\to 0}{\lim }{\left[{(1+\frac{\sin x-x}{x})}^{\frac{x}{\sin x-x}}\right]}^{\frac{3(\sin x-x)}{x^3}}=$$$$e^{\underset{x\to 0}{\lim }\left(\frac{3(\sin x-x)}{x^3}\right)}=e^{\underset{x\to 0}{\lim }\left(\frac{3(\cos x-1)}{3x^2}\right)}=$$$$e^{\underset{x\to 0}{\lim }\left(\frac{-\sin x}{2x}\right)}=e^{-\frac{1}{2}}=\frac{1}{\sqrt{e}}$$

Commented by mr W last updated on 05/Feb/20

$$correctis\frac{1}{e^{\frac{3}{6}}}=\frac{1}{\sqrt{e}}\approx 0.6065$$

Commented by john santu last updated on 05/Feb/20

$$ooyes$$

Commented by jagoll last updated on 05/Feb/20

$$thankyoumrWandjohn$$

Commented by abdomathmax last updated on 05/Feb/20

$$letf\left(x\right)={\left(\frac{sinx}{x}\right)}^{\frac{3}{x^2}}\Rightarrow f\left(x\right)=e^{\frac{3}{x^2}ln\left(\frac{sinx}{x}\right)}$$$$wehavesinx=x-\frac{x^3}{3!}+o\left(x^5\right)\Rightarrow$$$$\frac{sinx}{x}=1-\frac{x^2}{6}+o\left(x^4\right)\Rightarrow ln\left(\frac{sinx}{x}\right)$$$$=ln(1-\frac{x^2}{6}+o\left(x^4\right))\sim -\frac{x^2}{6}\Rightarrow \frac{3}{x^2}ln\left(\frac{sinx}{x}\right)\sim -\frac{1}{2}$$$$\Rightarrow {lim}_{x\to 0}f\left(x\right)=e^{-\frac{1}{2}}=\frac{1}{\sqrt{e}}$$

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