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Question Number 32517 by abdo imad last updated on 26/Mar/18

$${calculatelim}_{x\to 0^{+}}\frac{x^{sinx}-{\left(sinx\right)}^x}{x}.$$

Commented by abdo imad last updated on 27/Mar/18

$$wehave{x}^{sinx}=e^{sinxln\left(x\right)}butsinx\sim x-\frac{x^3}{6}$$$$e^{sinxln\left(x\right)}\sim e^{(x-\frac{x^3}{6})lnx}\sim 1+(x-\frac{x^3}{6})lnx$$$${\left(sinx\right)}^x=e^{xln\left(sinx\right)}\sim e^{xln(x-\frac{x^3}{3})}=e^{xlnx+xln(1-\frac{x^2}{3})}$$$$\sim e^{xlnx}{e}^{-\frac{x^3}{3}}\sim e^{xlnx}(1-\frac{x^3}{3})\sim \Rightarrow$$$$\frac{x^{sinx}-{\left(sinx\right)}^x}{x}\sim \frac{1+(x-\frac{x^3}{6})lnx-e^{xlnx}+\frac{x^3}{3}{e}^{xlnx}}{x}$$$$\sim \frac{x^2}{3}{e}_{x\to 0^{+}}^{xln\left(x\right)}\to 0.$$

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