Question

Consider the function $f\left(x\right)=\{\begin{array}{cc}{x}^2-5,& x\le 3\\ \sqrt{x+13},& x>3\end{array}$.

What is $\underset{x\to 3}{lim}f\left(x\right)$ equal to.

• A. $2$
• B. $4$
• C. $5$
• D. $13$

Answer 1

The correct option is B $4$

Given $f\left(x\right)=\{\begin{array}{cc}{x}^2-5,& x\le 3\\ \sqrt{x+13},& x>3\end{array}$.

$\underset{x\to 3^{-}}{lim}f\left(x\right)=\underset{x\to 3^{-}}{lim}(x^2-5)=9-5=4$ $[x\to 3^{-}\Rightarrow x<3\Rightarrow f(x)=x^2-5]$

$\underset{x\to 3^{+}}{lim}f\left(x\right)=\underset{x\to 3^{+}}{lim}\sqrt{x+13}=\sqrt{16}=4$ $[x\to 3^{+}\Rightarrow x>3\Rightarrow f(x)=\sqrt{x+13}]$

$\because$ LHL$=$RHL

$\Rightarrow \underset{x\to 3}{lim}f\left(x\right)=4$