Question

Find $\underset{x\to 3}{\lim }f\left(x\right)$, where $f\left(x\right)=\left\{\begin{array}{ll}4,& \mathrm{if}x>3\\ x+1,& \mathrm{if}x<3\end{array}\right.$

Answer 1

We have,

$f\left(x\right)=\left\{\begin{array}{ll}4,& \mathrm{if}x>3\\ x+1,& \mathrm{if}x<3\end{array}\right.$

LHL of f(x) at x = 3

= $\underset{x\to 3^{-}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(3-h\right)=\underset{h\to 0}{\lim }\left(3-h+1\right)=4$

RHL of f(x) at x = 3

= $\underset{x\to 3^{+}}{\lim }f\left(x\right)=\underset{h\to 0}{\lim }f\left(3+h\right)=\underset{h\to 0}{\lim }4=4$

Clearly, $\underset{x\to 3^{-}}{\lim }f\left(x\right)=\underset{x\to 3^{+}}{\lim }f\left(x\right)=4$

$\therefore \underset{x\to 3}{\lim }f\left(x\right)=4$