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I have a question regarding a three-variable limit.

I have to evaluate the following limit:

\begin{equation} \lim_{(x,y,z) \rightarrow (0,0,0)} \sin\left(\frac{1}{x}\right)\ \cos\left (\frac{1}{y}\right)\ \text{cos}(z) \end{equation}

I know that the limit does not exist. Now I have to show it. I was thinking of sequences $a_n = (x_n, y_n, z_n)$ and $b_n = (x_n, y_n, z_n)$ such that both have a different limiting value in this function. Problem is, I can not come up with such sequences. Can anybody help me out? Help is greatly appreciated!

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  • $\begingroup$ No sorry I made a stupid mistake. The sin$(z)$ was supposed to be a cos$(z)$ $\endgroup$
    – user444389
    Jun 30, 2017 at 15:57

2 Answers 2

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Hint. Take for example $(1/(\frac{\pi}{2}+2n\pi),1/(2n\pi),0)$ and $(1/(\pi/2+2n\pi),1/((2n+1)\pi),0)$.

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Let $$x_n=\frac {1}{\frac {\pi}{2}+2n\pi} $$

and $$z_n=\frac {1}{2n\pi} $$ then

$$f (x_n,x_n,z_n)=1$$ and $$f (z_n,z_n,z_n)=0$$ the limit doesn't exist.

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