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

{c a l c u l a t e l i m}_{x \to 0^{+}} \frac{x^{s i n x} - {(s i n x)}^{x}}{x} .

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

w e h a v e x^{s i n x} = e^{s i n x l n (x)} b u t s i n x \sim x - \frac{x^{3}}{6}

e^{s i n x l n (x)} \sim e^{(x - \frac{x^{3}}{6}) l n x} \sim 1 + (x - \frac{x^{3}}{6}) l n x

{(s i n x)}^{x} = e^{x l n (s i n x)} \sim e^{x l n (x - \frac{x^{3}}{3})} = e^{x l n x + x l n (1 - \frac{x^{2}}{3})}

\sim e^{x l n x} e^{- \frac{x^{3}}{3}} \sim e^{x l n x} (1 - \frac{x^{3}}{3}) \sim \Rightarrow

\frac{x^{s i n x} - {(s i n x)}^{x}}{x} \sim \frac{1 + (x - \frac{x^{3}}{6}) l n x - e^{x l n x} + \frac{x^{3}}{3} e^{x l n x}}{x}

\sim \frac{x^{2}}{3} e_{x \to 0^{+}}^{x l n (x)} \to 0 .