Find f(x) andf(x), where f(x) =

Question:

Find $\lim _{x \rightarrow 0} f(x)$ and $\lim _{x \rightarrow 1} f(x)$, where $f(x)= \begin{cases}2 x+3, & x \leq 0 \\ 3(x+1), & x>0\end{cases}$

Solution:

The given function is

$f(x)= \begin{cases}2 x+3, & x \leq 0 \\ 3(x+1), & x>0\end{cases}$

$\lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0}[2 x+3]=2(0)+3=3$

$\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0} 3(x+1)=3(0+1)=3$

$\therefore \lim _{x \rightarrow 0^{-}} f(x)=\lim _{x \rightarrow 0^{+}} f(x)=\lim _{x \rightarrow 0} f(x)=3$

$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1} 3(x+1)=3(1+1)=6$

$\lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1} 3(x+1)=3(1+1)=6$

$\therefore \lim _{x \rightarrow 1^{-}} f(x)=\lim _{x \rightarrow 1^{+}} f(x)=\lim _{x \rightarrow 1} f(x)=6$