Question

Determine the value of the constant k so that the function
$f\left(x\right)=\left\{\begin{array}{ll}\frac{x^2-3x+2}{x-1},\mathrm{if}& x\ne 1\\ k,\mathrm{if}& x=1\end{array}\mathrm{is}\mathrm{continuous}\mathrm{at}x=1\right.$

Answer 1

Given:
$f\left(x\right)=\left\{\begin{array}{l}\frac{x^2-3x+2}{x-1},\mathrm{if}x\ne 1\\ k,\mathrm{if}x=1\end{array}\right.$
If $f\left(x\right)$ is continuous at x = 1, then,
$\underset{\mathrm{x}\to 1}{\lim }f\left(x\right)=f\left(1\right)$
⇒ $\underset{\mathrm{x}\to 1}{\lim }\frac{x^2-3x+2}{x-1}=k$
⇒ $\underset{\mathrm{x}\to 1}{\lim }\frac{\left(x-2\right)\left(x-1\right)}{x-1}=k$
⇒ $\underset{\mathrm{x}\to 1}{\lim }\left(x-2\right)=k$
⇒ $k=-1$