Question

Find the values of a so that the function
$f\left(x\right)=\left\{\begin{array}{ll}ax+5,\mathrm{if}& x\le 2\\ x-1,\mathrm{if}& x>2\end{array}\right.\mathrm{is}\mathrm{continuous}\mathrm{at}x=2.$

Answer 1

Given:
$f\left(x\right)=\left\{\begin{array}{l}ax+5,\mathrm{if}x\le 2\\ x-1,\mathrm{if}x>2\end{array}\right.$

We observe
(LHL at x = 2) = $\underset{\mathrm{x}\to 2^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(2-h\right)$ = ​$\underset{h\to 0}{\lim }a\left(2-h\right)+5=2a+5$

(RHL at x = 2) = $\underset{\mathrm{x}\to 2^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(2+h\right)$ = ​$\underset{h\to 0}{\lim }\left(2+h-1\right)$ = $1$

And, $f\left(2\right)=a\left(2\right)+5=2a+5$

Since $f\left(x\right)$ is continuous at x = 2, we have
$\underset{\mathrm{x}\to 2^{-}}{\lim }f\left(x\right)=\underset{\mathrm{x}\to 2^{+}}{\lim }f\left(x\right)=f\left(2\right)$
⇒ $2a+5=1$
⇒ $2a=-4$
⇒ $a=-2$