Question:
Evaluate the following limits:
$\lim _{x \rightarrow 0} \frac{\tan (x / 2)}{3 x}$
Solution:
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: $\lim _{x \rightarrow 0} \frac{\operatorname{tanx}}{x}=1$
So, by using the above formula, we have
$\lim _{x \rightarrow 0} \frac{\tan (x / 2)}{3 x}=\lim _{x \rightarrow 0} \frac{\tan (x / 2)}{6(x / 2)}=\frac{1}{6}$ [Divide and multiply with 2 on denominator]
Therefore, $\lim _{x \rightarrow 0} \frac{\tan (x / 2)}{3 x}=\frac{1}{6}$