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Question Number 59694 by meme last updated on 13/May/19

$$li\underset{x\to 0}{m}\frac{cos(\sqrt{\mid x\mid )-1}}{x}=?$$

Commented by maxmathsup by imad last updated on 13/May/19

$$letA\left(x\right)=\frac{cos\left(\sqrt{\mid x\mid }\right)-1}{x}wehavecos\left(u\right)\sim 1-\frac{u^2}{2}(u\in V\left(0\right))\Rightarrow$$$$cos\left(\sqrt{\mid x\mid }\right)\sim 1-\frac{\mid x\mid }{2}\Rightarrow cos\left(\sqrt{\mid x\mid }\right)-1\sim -\frac{\mid x\mid }{2}nowitsclearthat$$$${lim}_{x\to 0^{+}}A\left(x\right)=-\frac{1}{2}and{lim}_{x\to 0^{-}}A\left(x\right)=\frac{1}{2}$$

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