# Limit with three variables

I have a question regarding a three-variable limit.

I have to evaluate the following limit:

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

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!

• No sorry I made a stupid mistake. The sin$(z)$ was supposed to be a cos$(z)$ – user444389 Jun 30 '17 at 15:57

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