Find the value of k for which the function


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 $x=2$



$f(x)=\left\{\begin{array}{cc}\frac{x^{2}+3 x-10}{x-2}, & x \neq 2 \\ k, & x=2\end{array}\right.$

$\lim _{x \rightarrow 2^{-}}\left(\frac{x^{2}+3 x-10}{x-2}\right)=\lim _{x \rightarrow 2^{-}}(x+5)=7$


$\lim _{x \rightarrow 2^{+}}\left(\frac{x^{2}+3 x-10}{x-2}\right)=\lim _{x \rightarrow 2^{+}}(x+5)=7$

If $f(x)$ is continuous at $x=2$, then

$\lim _{x \rightarrow 2^{-}} f(x)=f(2)=\lim _{x \rightarrow 2^{+}} f(x)$

$\Rightarrow k=7$

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