Evaluate the following limits:

To Find: Limits

NOTE: First Check the form of imit. Used this method if the limit is satisfying any one from 7 indeterminate form.

In this Case, indeterminate Form is $\frac{0}{0}$

Formula used: $\lim _{x \rightarrow 0} \frac{1-\cos x}{x^{2}}=\frac{1}{2}$ and $\lim _{x \rightarrow 0} \frac{\tan x}{x}=1$

Divide numerator and denominator by $x^{2}$, we have

So, by using the above formula, we have

$\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{3 \tan ^{2} x}=\lim _{x \rightarrow 0} \frac{\frac{4[1-\cos 2 x]}{(4) x^{2}}}{\frac{3 \tan ^{2} x}{x^{2}}}=\frac{4}{6}=\frac{2}{3}$

Therefore, $\lim _{x \rightarrow 0} \frac{1-\cos 2 x}{3 \tan ^{2} x}=\frac{1}{6}$