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Question Number 145791 by mathmax by abdo last updated on 08/Jul/21

$$\mathrm{find}{\lim }_{\mathrm{x}\to 0}{\int }_0^{\mathrm{x}}\frac{{\mathrm{e}}^{\mathrm{t}}+{\mathrm{e}}^{-\mathrm{t}}-2}{1-\mathrm{cosx}}\mathrm{dx}$$

Commented by mathmax by abdo last updated on 09/Jul/21

$${\lim }_{\mathrm{x}\to 0}{\int }_0^{\mathrm{x}}\frac{{\mathrm{e}}^{\mathrm{t}}+{\mathrm{e}}^{-\mathrm{t}}-2}{1-\mathrm{cosx}}\mathrm{dt}$$

Commented by mathmax by abdo last updated on 10/Jul/21

$$\mathrm{let}\mathrm{use}\mathrm{hospital}\mathrm{rule}$$$$={\lim }_{\mathrm{x}\to 0}\frac{1}{1-\mathrm{cosx}}({\mathrm{e}}^{\mathrm{x}}+{\mathrm{e}}^{-\mathrm{x}}-2)={\lim }_{\mathrm{x}\to 0}\frac{{\mathrm{e}}^{\mathrm{x}}-{\mathrm{e}}^{-\mathrm{x}}}{\mathrm{sinx}}$$$$={\lim }_{\mathrm{x}\to 0}\frac{(1+\mathrm{x}+\frac{{\mathrm{x}}^2}{2})-(1-\mathrm{x}+\frac{{\mathrm{x}}^2}{2})}{\mathrm{sinx}}={\lim }_{\mathrm{x}\to 0}\frac{2\mathrm{x}}{\mathrm{sinx}}=2$$

Answered by Olaf_Thorendsen last updated on 09/Jul/21

$$f\left(x\right)={\int }_0^x\frac{e^t+e^{-t}-2}{1-\cos x}dx$$$$f\left(x\right)={\left[\frac{2\mathrm{sh}t-2t}{1-\cos x}\right]}_0^x$$$$f\left(x\right)=\frac{2\mathrm{sh}x-2x}{1-\cos x}$$$$f\left(x\right)\underset{0}{\sim }\frac{2(x+\frac{x^3}{3!})-2x}{1-(1-\frac{x^2}{2!})}\underset{0}{\sim }\frac{x}{3}\to 0$$

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