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Question Number 31506 by abdo imad last updated on 09/Mar/18

$$letf\left(x\right)={\int }_x^{2x}\frac{sht}{t}dt$$$$1)calculate{f}^{{}^{\prime }}\left(x\right)$$$$2)find{lim}_{x\to 0}f\left(x\right).$$

Commented by abdo imad last updated on 12/Mar/18

$$1)f^{{}^{\prime }}\left(x\right)=2\frac{sh\left(2x\right)}{2x}-\frac{shx}{x}=\frac{sh\left(2x\right)-shx}{x}.$$$$2)\mathrm{\exists }c\in ]x,2x[/f\left(x\right)=shc{\int }_x^{2x}\frac{dt}{t}=sh\left(c\right)ln{\left(2\right)}_{x\to 0}\to 0$$

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