How can I find $\delta$ given $\epsilon$?

Question

How do I need to show that there exist $\delta>0$ corresponding to $\epsilon= 0.03$, when $(x,y,z)\to (0,0,0)$ using $\epsilon$-$\delta$ definition of limit. Given that $f(x,y,z)$ is $$f(x,y,z)=\tan^2 x + \tan^2 y + \tan^2 z.$$

Answer 1

Hint:

This plot of $\dfrac{\tan x}x$ shows you that you can ensure $c_0|t|\le|\tan t|\le c_1|t|$ in your favorite range around $0$ ($c_0=1,c_1=\tan x_{\max}/x_{\max}$). With this you can easily bound.