Question

Find $li{m}_{x\to 3}f\left(x\right)$, where

$f\left(x\right)=\{\begin{array}{c}4,\text{if}x>3\\ x+1,\text{if}x<3\end{array}$

Answer 1

$li{m}_{x\to 3^{+}}f\left(x\right)=4$

Now, $li{m}_{x\to 3^{-}}f\left(x\right)=li{m}_{x\to 3^{-}}(x+1)$

Put x =3 -h

$\Rightarrow h=3-x$

as $x\to 3^{-}\Rightarrow x<3$ slightly

$\Rightarrow h\to 0^{+}$

$li{m}_{x\to 0^{+}}(3-h+1)=3+1=4$

Since, $li{m}_{x\to 3^{+}}f\left(x\right)=4=li{m}_{x\to 3^{-}}f\left(x\right)$

$\therefore li{m}_{x\to 3}f\left(x\right)=4.$