Chapter 11.2, Problem 54E

To determine

To calculate : The limit, $\underset{x\to 1}{\lim }f\left(x\right)$ by finding one-sided limits of the function, f(x). Also, draw the graph of the function such that $f\left(x\right)=\{\begin{array}{c}2x+1,x<1\\ x+2,x\ge 1\end{array}$ .

Answer to Problem 54E

The value of $\underset{x\to 1}{\lim }f\left(x\right)$ is 3. The graph of the function is given by,

  

Explanation of Solution

Given information : The function is $f\left(x\right)=\{\begin{array}{c}2x+1,x<1\\ x+2,x\ge 1\end{array}$ .

Concept 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 $x<1$ , f(x) is given by $2x+1$ .

When $x\ge 1$ , f(x) is given by $x+2$ .

Draw the graph of the function f(x).

  

For left hand limit, substitute $x=1$ in the function, $f\left(x\right)=2x+1$ .

  $\begin{array}{c}\underset{x\to 1^{-}}{\lim }f\left(x\right)=\underset{x\to 1^{-}}{\lim }\left(2x+1\right)\\ =2\left(1\right)+1\\ =3\end{array}$

For right hand limit, substitute $x=1$ in the function, $f\left(x\right)=x+2$ .

  $\begin{array}{c}\underset{x\to 1^{+}}{\lim }f\left(x\right)=\underset{x\to 1^{+}}{\lim }\left(x+2\right)\\ =\left(1\right)+2\\ =3\end{array}$

As the left hand limit and right hand limit are equal, the value of $\underset{x\to 1}{\lim }f\left(x\right)$ is 3.