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Question Number 40095 by maxmathsup by imad last updated on 15/Jul/18

$$letf\left(x\right)=cos\left(x\right)cos\left(\frac{1}{x}\right)isfhavealimitatpoint0?$$$$$$

Answered by math khazana by abdo last updated on 26/Jul/18

$$let{x}_n=\frac{2}{n\pi }wehave{lim}_{n\to +\mathrm{\infty }}{x}_n=0and$$$$f\left(x_n\right)=cos\left(\frac{2}{n\pi }\right)cos\left(\frac{n\pi }{2}\right)\Rightarrow$$$$f\left(x_{2n}\right)=cos\left(\frac{2}{2n\pi }\right)cos\left(\frac{2n\pi }{2}\right)=cos\left(\frac{1}{n\pi }\right){(-1)}^n$$$$and{(-1)}^{n}donthaveanylimitandthatprove$$$$thatfhaventanylimitatpoint0$$

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