Question

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

Answer 1

$$\begin{gathered} f\left(x\right)=\left\{\begin{array}{ll}2x+3,& x\le 0\\ 3\left(x+1\right),& x>0\end{array}\right. \\ \mathrm{LHL}: \\ \underset{x\to 0^{-}}{\lim }f\left(x\right) \\ =\underset{x\to 0^{-}}{\lim }\left(2x+3\right) \\ \mathrm{Let}x=0-h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }\left[2\left(0-h\right)+3\right] \\ =\underset{h\to 0}{\lim }\left[-2h+3\right] \\ =3 \\ \mathrm{RHL}: \\ \underset{x\to 0^{+}}{\lim }f\left(x\right) \\ \underset{x\to 0^{+}}{\lim }3\left(x+1\right) \\ \mathrm{Let}x=0+h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }3\left(0+h+1\right) \\ =\underset{h\to 0}{\lim }\left(3h+3\right) \\ =3 \\ \therefore \mathrm{LHL}=\mathrm{RHL} \\ \therefore \underset{x\to 0}{\lim }f\left(x\right)=3 \end{gathered}$$

(ii)
$$\begin{gathered} \underset{x\to 1}{\lim }f\left(x\right)=? \\ \mathrm{LHL}: \\ \underset{x\to 1^{-}}{\lim }f\left(x\right) \\ =\underset{x\to 1^{-}}{\lim }3\left(x+1\right) \\ \mathrm{Let}x=1-h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }3\left(1-h+1\right) \\ =6 \\ \mathrm{RHL}: \\ \underset{x\to 1^{+}}{\lim }f\left(x\right) \\ =\underset{x\to 1^{+}}{\lim }3\left(x+1\right) \\ \mathrm{Let}x=1+h,\mathrm{where}h\to 0. \\ \underset{h\to 0}{\lim }3\left(1+h+1\right) \\ =6 \\ \therefore \mathrm{LHL}=\mathrm{RHL} \\ \underset{x\to 1}{\lim }f\left(x\right)=6 \end{gathered}$$