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Question Number 166560 by mathlove last updated on 22/Feb/22

Answered by qaz last updated on 22/Feb/22

$$\underset{\mathrm{x}\to 0}{\lim }\frac{\ln ({\mathrm{e}}^{\sin \mathrm{x}}+{\mathrm{x}}^3)-\sin \mathrm{x}}{\sin \mathrm{x}-\mathrm{x}\sqrt{1-{\mathrm{x}}^2}}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{\ln \frac{{\mathrm{e}}^{\sin \mathrm{x}}+{\mathrm{x}}^3}{{\mathrm{e}}^{\sin \mathrm{x}}}}{\mathrm{x}-\frac{1}{6}{\mathrm{x}}^3-\mathrm{x}(1-\frac{1}{2}{\mathrm{x}}^2)+\mathrm{o}\left({\mathrm{x}}^3\right)}$$$$=\underset{\mathrm{x}\to 0}{\lim }\frac{\frac{{\mathrm{e}}^{\sin \mathrm{x}}+{\mathrm{x}}^3}{{\mathrm{e}}^{\sin \mathrm{x}}}-1}{\frac{1}{3}{\mathrm{x}}^3}$$$$=3$$

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