Question

Find the value of k for which the function $f\left(x\right)=\left\{\begin{array}{l}\begin{array}{cc}\frac{x^2+3x-10}{x-2},& x\ne 2\\ k,& x=2\end{array}\end{array}\right.$is continuous at x = 2

Answer 1

Given,
$f\left(x\right)=\left\{\begin{array}{l}\begin{array}{cc}\frac{x^2+3x-10}{x-2},& x\ne 2\\ k,& x=2\end{array}\end{array}\right.$
$$\begin{gathered} \underset{x\to 2^{-}}{\lim }\left(\frac{x^2+3x-10}{x-2}\right)=\underset{x\to 2^{-}}{\lim }\left(x+5\right)=7 \\ f\left(2\right)=k \\ \underset{x\to 2^{+}}{\lim }\left(\frac{x^2+3x-10}{x-2}\right)=\underset{x\to 2^{+}}{\lim }\left(x+5\right)=7 \end{gathered}$$
If $f\left(x\right)$ is continuous at $x=2$, then
$\underset{x\to 2^{-}}{\lim }f\left(x\right)=f\left(2\right)=\underset{x\to 2^{+}}{\lim }f\left(x\right)$
$\Rightarrow k=7$