[tex]\displaystyle\lim_{x\to0}\frac{\tan x^2}x=\lim_{x\to0}x\frac{\tan x^2}{x^2}=\lim_{x\to0}\frac x{\cos x^2}\times\lim_{x\to0}\frac{\sin x^2}{x^2}[/tex]
Recall that
[tex]\displaystyle\lim_{x\to0}\frac{\sin x}x=1[/tex]
and so replacing [tex]x[/tex] with [tex]x^2[/tex], you get that the second limit is 1. Meanwhile,
[tex]\displaystyle\lim_{x\to0}\dfrac x{\cos x^2}=\dfrac01=0[/tex]
so the limit is 0.
