Question:
Find the value of $k$ for which the function $f(x)=\left\{\begin{array}{cl}\frac{x^{2}+3 x-10}{x-2}, & x \neq 2 \\ k, & x=2\end{array}\right.$ is continuous at $x=2$
Solution:
Given,
$f(x)=\left\{\begin{array}{cc}\frac{x^{2}+3 x-10}{x-2}, & x \neq 2 \\ k, & x=2\end{array}\right.$
$\lim _{x \rightarrow 2^{-}}\left(\frac{x^{2}+3 x-10}{x-2}\right)=\lim _{x \rightarrow 2^{-}}(x+5)=7$
$f(2)=k$
$\lim _{x \rightarrow 2^{+}}\left(\frac{x^{2}+3 x-10}{x-2}\right)=\lim _{x \rightarrow 2^{+}}(x+5)=7$
If $f(x)$ is continuous at $x=2$, then
$\lim _{x \rightarrow 2^{-}} f(x)=f(2)=\lim _{x \rightarrow 2^{+}} f(x)$
$\Rightarrow k=7$