Question

For what value of k is the following function continuous at x = 1?
$f\left(x\right)=\left\{\begin{array}{ll}\frac{x^2-1}{x-1},& x\ne 1\\ k,& x=1\end{array}\right.$

Answer 1

Given: $f\left(x\right)=\left\{\begin{array}{l}\frac{x^2-1}{x-1},x\ne 1\\ k,x=1\end{array}\right.$
If $f\left(x\right)$ is continuous at x = 1, then
$\underset{\mathrm{x}\to 1}{\lim }f\left(x\right)=f\left(1\right)$
⇒ $\underset{\mathrm{x}\to 1}{\lim }\frac{x^2-1}{x-1}=k$
⇒ $\underset{\mathrm{x}\to 1}{\lim }\frac{\left(x-1\right)\left(x+1\right)}{x-1}=k$
⇒ $\underset{\mathrm{x}\to 1}{\lim }\left(x+1\right)=k$
⇒ $k=2$