Chapter 12.2, Problem 56E

(a)..

To determine

To find: The limit for the given graph.

Answer to Problem 56E

The limit of the function is $2$ .

Explanation of Solution

Given:

The given function is,

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

The given diagram is shown in Figure 1

  

Figure 1

The given limit is $\underset{x\to 0}{\text{lim}}f\left(x\right)$ .

Calculation:

For the value of $x<0$ , $f\left(x\right)$ is given by $2x$ and use the direct substitution as,

Consider the limit of the given function is obtained as,

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

Thus, the limit of the function is $2$ .

(b)..

To determine

To find: The limit for the given graph.

Answer to Problem 56E

The limit of the function is $1$ .

Explanation of Solution

Given:

The given limit is $\underset{x\to 0^{+}}{\text{lim}}f\left(x\right)$ .

Calculation:

Consider the limit of the given function is obtained as,

  $\begin{array}{c}\underset{x\to 0^{+}}{\text{lim}}f\left(x\right)=\underset{x\to 0^{+}}{\text{lim}}\left(x^2+1\right)\\ =0^2+1\\ =1\end{array}$

Thus, the limit of the function is $1$ .

(c)..

To determine

To find: The limit for the given graph.

Answer to Problem 56E

The limit of the function is $2$ .

Explanation of Solution

Given:

The given limit is $\underset{x\to 0}{\text{lim}}f\left(x\right)$ .

Calculation:

Consider the limit of the given function is obtained as,

  $\begin{array}{c}\underset{x\to 0}{\text{lim}}f\left(x\right)=\underset{x\to 0}{\text{lim}}\left(2x\right)\\ =2\cdot 0\\ =2\end{array}$

Thus, the limit of the function is $2$ .