Question Number 31506 by abdo imad last updated on 09/Mar/18
$${let}\:{f}\left({x}\right)=\int_{{x}} ^{\mathrm{2}{x}} \:\frac{{sht}}{{t}}{dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{f}^{'} \left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{find}\:{lim}_{{x}\rightarrow\mathrm{0}} \:{f}\left({x}\right)\:. \\ $$
Commented by abdo imad last updated on 12/Mar/18
![1)f^′ (x)=2 ((sh(2x))/(2x)) −((shx)/x) = ((sh(2x)−shx)/x) . 2) ∃ c ∈]x,2x[ /f(x)=sh c ∫_x ^(2x) (dt/t)=sh(c)ln(2)_(x→0) →0](Q31700.png)
$$\left.\mathrm{1}\right){f}^{'} \left({x}\right)=\mathrm{2}\:\frac{{sh}\left(\mathrm{2}{x}\right)}{\mathrm{2}{x}}\:−\frac{{shx}}{{x}}\:=\:\frac{{sh}\left(\mathrm{2}{x}\right)−{shx}}{{x}}\:. \\ $$$$\left.\mathrm{2}\left.\right)\:\exists\:{c}\:\in\right]{x},\mathrm{2}{x}\left[\:/{f}\left({x}\right)={sh}\:{c}\:\int_{{x}} ^{\mathrm{2}{x}} \:\frac{{dt}}{{t}}={sh}\left({c}\right){ln}\left(\mathrm{2}\right)_{{x}\rightarrow\mathrm{0}} \:\rightarrow\mathrm{0}\right. \\ $$