Chapter 11.2, Problem 56E

To determine

To Calculate : The limit, $\underset{x\to 0}{\lim }f\left(x\right)$ by finding one-sided limits of the function, f(x). Also, draw the graph of the function such that $f\left(x\right)=\{\begin{array}{c}{x}^2+3,x\le 0\\ x+5,x>0\end{array}$ .

Answer to Problem 56E

The limit, $\underset{x\to 0}{\lim }f\left(x\right)$ does not exists . The graph of the function is given by,

  

Explanation of Solution

Given information : The function is $f\left(x\right)=\{\begin{array}{c}{x}^2+3,x\le 0\\ x+5,x>0\end{array}$ .

Formula used : If the left hand limit and right hand limits are equal then $\underset{x\to x}{\lim }f\left(x\right)=L$ otherwise the limits does not exists.

Calculation : When $x\le 0$ , f(x) is given by $x^2+3$ .

When $x>0$ , f(x) is given by $x+5$ .

Draw the graph of the function f(x).

  

For left hand limit, substitute $x=0$ in the function, $f\left(x\right)=x^2+3$ .

  $\begin{array}{c}\underset{x\to 0^{-}}{\lim }f\left(x\right)=\underset{x\to 0^{-}}{\lim }\left(x^2+3\right)\\ ={\left(0\right)}^2+3\\ =3\end{array}$

For right hand limit, substitute $x=0$ in the function, $f\left(x\right)=x+5$ .

  $\begin{array}{c}\underset{x\to 0^{+}}{\lim }f\left(x\right)=\underset{x\to 0^{+}}{\lim }\left(x+5\right)\\ =\left(0\right)+5\\ =5\end{array}$

As the left hand limit and right hand limit are not equal, the limit does not exists.