Question

Find the values of a and b so that the function defined by f(x) = $\{\begin{array}{c}ax+1,ifx\le 3\\ bx+3,ifx>3\end{array}$ is continuous at x=3.

Answer 1

Here, f(x) $\{\begin{array}{c}ax+1,ifx\le 3\\ bx+3,ifx>3\end{array}$

LHL = $\underset{x\to 3^{-}}{lim}$ f(x) = $\underset{x\to 3^{-}}{lim}$ (ax+1)

Putting x=3-h as $x\to 3^{-}$ when $h\to 0$
$\therefore$ $\underset{h\to 0}{lim}$ [a(3-h)+1] $=\underset{h\to 0}{lim}(3a-ah+1)=3a+1$

RHL = $\underset{x\to 3^{+}}{lim}$ f(x) = $\underset{x\to 3^{+}}{lim}$ (bx+3)

Putting x=3-h as $x\to 3^{+}$ when $h\to 0$

$\therefore$ $\underset{h\to 0}{lim}$ [b(3-h)+3] $\underset{h\to 0}{lim}$ (3b-bh+3)=3b+3
Also f(3)=3a+1 $\therefore f\left(x\right)=ax+1$

Since, f(x) is continuous at x=3.

LHL=RHL=f(3)

$\Rightarrow 3a+1=3b+3\Rightarrow 3a=3b+2\Rightarrow a=b+\frac{2}{3}$