Question

Find the relationship between 'a' and 'b' so that the function 'f' defined by
$f\left(x\right)=\left\{\begin{array}{ll}ax+1,& \mathrm{if}x\le 3\\ bx+3,& \mathrm{if}x>3\end{array}\right.$
is continuous at x = 3.

Answer 1

Given: $f\left(x\right)=\left\{\begin{array}{l}ax+1,\mathrm{if}x\le 3\\ bx+3,\mathrm{if}x>3\end{array}\right.$

We have
(LHL at x = 3) = $\underset{x\to 3^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(3-h\right)=\underset{h\to 0}{\lim }a\left(3-h\right)+1=3a+1$

(RHL at x = 3) = $\underset{x\to 3^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(3+h\right)=\underset{h\to 0}{\lim }b\left(3+h\right)+3=3b+3$

$$\begin{gathered} \mathrm{If}f\left(x\right)\mathrm{is}\mathrm{continuous}\mathrm{at}x=3,\mathrm{then} \\ \underset{\mathrm{x}\to 3^{-}}{\lim }f\left(\mathrm{x}\right)=\underset{\mathrm{x}\to 3^{+}}{\lim }f\left(\mathrm{x}\right) \\ \Rightarrow 3a+1=3b+3 \\ \Rightarrow 3a-3b=2 \end{gathered}$$

Hence, the required relationship between $a\&b\mathrm{is}3a-3b=2$.