Question

If $f\left(x\right)=\{\begin{array}{cc}2x+3& ,x\le 0\\ 3(x+1)& ,x>0\end{array}$ Find $li{m}_{x\to 0}f\left(x\right)$ and $li{m}_{x\to 1}f\left(x\right).$

Answer 1

$f\left(x\right)=\{\begin{array}{cc}2x+3& ,x\le 0\\ 3(x+1)& ,x>0\end{array}$

LHL:

$li{m}_{x\to 0}f\left(x\right)=li{m}_{x\to }(2x+3)$

Let x =0 -h, h =-x

as $x\to 0^{-}\Rightarrow x<0$ (slightly)

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

$li{m}_{x\to 0^{+}}\left[2\right(0-h)+3]$

$=li{m}_{x\to 0^{+}}[-2h+3]=3$

RHL:

LHL:

$li{m}_{x\to 0}f\left(x\right)=li{m}_{x\to }(2x+3)$

Let x =0 -h, h =-x

as $x\to 0^{-}\Rightarrow x<0$ (slightly)

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

$li{m}_{x\to 0^{+}}\left[2\right(0-h)+3]$

$=li{m}_{x\to 0^{+}}[-2h+3]=3$

RHL:

$li{m}_{x\to 0^{+}}f\left(x\right)$

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

Let $x=0+h,\Rightarrow h=x$

as $x\to 0^{+}$ (slightly)

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

$li{m}_{x\to 0^{+}}(3h+3)=2$

$\therefore LHL=RHL$

$li{m}_{x\to 1^{+}}f\left(x\right)$

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

Let x =1+h, where $h\to 0^{+}$

$li{m}_{x\to 0}3(1+h+1)=6$

$\therefore LHL=RHL$

$li{m}_{x\to 1}f\left(x\right)=6$