Question

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

Answer 1

The function $f\left(x\right)$ is given to be continuous at $x=2$.

Then $\underset{x\to 2}{lim}f\left(x\right)=f\left(2\right)$

or, $\underset{x\to 2}{lim}\frac{x^2+3x-10}{x-2}=k$ [Since $f\left(2\right)=k$]

or, $\underset{x\to 2}{lim}\frac{(x+5)(x-2)}{x-2}=k$

or, $\underset{x\to 2}{lim}(x+5)=k$

or, $k=7$.