Question:
For what value of k is the following function continuous at x = 1?
$f(x)=\left\{\begin{array}{rr}\frac{x^{2}-1}{x-1}, & x \neq 1 \\ k & , x=1\end{array}\right.$
Solution:
Given: $f(x)=\left\{\begin{array}{l}\frac{x^{2}-1}{x-1}, \quad x \neq 1 \\ k, \quad x=1\end{array}\right.$
If $f(x)$ is continuous at $x=1$, then
$\lim _{x \rightarrow 1} f(x)=f(1)$
$\Rightarrow \lim _{\mathrm{x} \rightarrow 1} \frac{x^{2}-1}{x-1}=k$
$\Rightarrow \lim _{\mathrm{x} \rightarrow 1} \frac{(x-1)(x+1)}{x-1}=k$
$\Rightarrow \lim _{\mathrm{x} \rightarrow 1}(x+1)=k$
$\Rightarrow k=2$