Question
Evaluate the left hand and right hand limits of the following function at $x=2$. Does $\lim _{x \rightarrow 2} f(x)$ exist ?
$
f(x)=\left\{\begin{array}{cll}
2 x+3, & \text { if } & x \leq 2 \\
x+5 & \text { if } & x>2
\end{array}\right.
$

Answer

$\begin{aligned} \text { L.H.L } & =\lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{-}}(2 x+3) \\ & =4+3 \\ & =7\end{aligned}$
and
$\begin{aligned} \text { R.H.L } & =\lim _{x \rightarrow 2} f(x)=\lim _{x \rightarrow 2^{+}}(x+5) \\ & =2+5 \\ & =7\end{aligned}$
Since $\quad \lim _{x \rightarrow 2^{-}} f(x)=\lim _{x \rightarrow 2^{+}} f(x)=7$
$\therefore \lim _{x \rightarrow 2} f(x)$ exists and is equal to 7.

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