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Question Number 85057 by kushdasbaghar@gmail.com last updated on 18/Mar/20

$$find\underset{x\to 0}{\lim ln}(\sin x\cos \frac{1}{x}+1)ifitexits.$$

Commented by abdomathmax last updated on 19/Mar/20

$$wehaveln(1+sinxcos\left(\frac{1}{x}\right))\sim sinxcos\left(\frac{1}{x}\right)(x\to 0)$$$$sinxcos\left(\frac{1}{x}\right)\sim xcos\left(\frac{1}{x}\right)and\mid xcos\left(\frac{1}{x}\right)\mid ⩽\mid x\mid \to 0\Rightarrow$$$${lim}_{x\to 0}sinxcos\left(\frac{1}{x}\right)=0\Rightarrow {lim}_{x\to 0}ln(sinxcos\left(\frac{1}{x}\right)+1)=0$$

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