Evaluate the Given limit:

$\lim _{x \rightarrow-2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}$

At $x=-2$, the value of the given function takes the form $\frac{0}{0}$

Now, $\lim _{x \rightarrow-2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}=\lim _{x \rightarrow-2} \frac{\left(\frac{2+x}{2 x}\right)}{x+2}$

$=\lim _{x \rightarrow-2} \frac{1}{2 x}$

$=\frac{1}{2(-2)}=\frac{-1}{4}$