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Question Number 160793 by cortano last updated on 06/Dec/21

$$\underset{x\to 0}{\lim }\frac{2^{\cos \mathrm{x}}-2}{{\mathrm{x}}^2}=?$$

Answered by qaz last updated on 06/Dec/21

$$\underset{\mathrm{x}\to 0}{\lim }\frac{2^{\cos \mathrm{x}}-2}{{\mathrm{x}}^2}=\underset{\mathrm{x}\to 0,\xi \to 1}{\lim }\frac{(\cos \mathrm{x}-1)2^{\xi }\ln 2}{{\mathrm{x}}^2}=-\ln 2$$

Answered by mr W last updated on 06/Dec/21

$$=\underset{x\to 0}{\lim }\frac{-\sin x{2}^{\cos x}\ln 2}{2x}$$$$=-\frac{2\times \ln 2}{2}$$$$=-\ln 2$$

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