Question:
Determine the value of the constant k so that the function
$f(x)=\left\{\begin{array}{rlr}\frac{x^{2}-3 x+2}{x-1}, & \text { if } & x \neq 1 \\ k & , & \text { if } x=1\end{array}\right.$ is continuous at $x=1$
Solution:
Given:
$f(x)=\left\{\begin{array}{l}\frac{x^{2}-3 x+2}{x-1}, \text { if } x \neq 1 \\ k, \text { if } x=1\end{array}\right.$
If $f(x)$ is continuous at $x=1$, then,
$\lim _{x \rightarrow 1} f(x)=f(1)$
$\Rightarrow \lim _{\mathrm{x} \rightarrow 1} \frac{x^{2}-3 x+2}{x-1}=k$
$\Rightarrow \lim _{\mathrm{x} \rightarrow 1} \frac{(x-2)(x-1)}{x-1}=k$
$\Rightarrow \lim _{\mathrm{x} \rightarrow 1}(x-2)=k$
$\Rightarrow k=-1$