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Question Number 160747 by amin96 last updated on 05/Dec/21

Commented by amin96 last updated on 05/Dec/21

$$$$does anyone have a solution?

Answered by qaz last updated on 06/Dec/21

$$\underset{\mathrm{x}\to 0}{\lim }\frac{(\mathrm{x}\cdot \mathrm{ch}\mathrm{x}-\cos \mathrm{x}){\mathrm{e}}^{\mathrm{x}}+1}{{\mathrm{x}}^3}-\frac{1}{\mathrm{x}}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{(\mathrm{xcosh}\mathrm{x}-\cos \mathrm{x}){\mathrm{e}}^{\mathrm{x}}+1-{\mathrm{x}}^2}{{\mathrm{x}}^3}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{(-1+\mathrm{x}+\frac{{\mathrm{x}}^2}{2}+\frac{{\mathrm{x}}^3}{2})(1+\mathrm{x}+\frac{{\mathrm{x}}^2}{2}+\frac{{\mathrm{x}}^3}{6})+1-{\mathrm{x}}^2+\mathrm{o}\left({\mathrm{x}}^3\right)}{{\mathrm{x}}^3}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{\frac{{4\mathrm{x}}^3}{3}+\mathrm{o}\left({\mathrm{x}}^3\right)}{{\mathrm{x}}^3}$$$$=\frac{4}{3}$$

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