Chapter 11.2, Problem 88E

(a)

To determine

To find: the limit of each function. If limit not exist then explain.

Answer to Problem 88E

  $\underset{x\to 0^{-}}{\lim }f\left(x\right)=0$

Explanation of Solution

Given:

  $\underset{x\to 0^{-}}{\lim }f\left(x\right)$

Calculation:

The limit is determined by $2x$ . As x approaches 0 from the left, $f\left(x\right)$ approaches 0 since $2\left(0\right)=0$ :

  $\underset{x\to 0^{-}}{\lim }f\left(x\right)=0$

Conclusion:

Therefore, $\underset{x\to 0^{-}}{\lim }f\left(x\right)=0$ .

(b)

To determine

To find: the limit of each function. If limit not exist then explain.

Answer to Problem 88E

  $\underset{x\to 0^{+}}{\lim }f\left(x\right)=0$

Explanation of Solution

Given:

  $\underset{x\to 0^{+}}{\lim }f\left(x\right)$

Calculation:

The limit is determined by $x^2+1$ . As x approaches 0 from the right, $f\left(x\right)$ approaches 1 since $0^2+1=1$ :

  $\underset{x\to 0^{+}}{\lim }f\left(x\right)=0$

Conclusion:

Therefore, $\underset{x\to 0^{+}}{\lim }f\left(x\right)=0$ .

(c)

To determine

To find: the limit of each function. If limit not exist then explain.

Answer to Problem 88E

  $\underset{x\to 0}{\lim }f\left(x\right)$ does not exist.

Explanation of Solution

Given:

  $\underset{x\to 0}{\lim }f\left(x\right)$

Calculation:

Since $\underset{x\to 0^{-}}{\lim }f\left(x\right)\ne \underset{x\to 0^{+}}{\lim }f\left(x\right)$ , then

  $\underset{x\to 0}{\lim }f\left(x\right)$ does not exist.

Conclusion:

Therefore, $\underset{x\to 0}{\lim }f\left(x\right)$ does not exist.