Chapter 11.1, Problem 51E

(a)

To determine

To evaluate: the given limit.

Answer to Problem 51E

Here, the limit of $\underset{x\to c}{\lim }\left[-2g\left(x\right)\right]$ is $-16$

Explanation of Solution

Given: $\underset{x\to c}{\lim }f\left(x\right)=4$

  $\underset{x\to c}{\lim }g\left(x\right)=8$

  $\underset{x\to c}{\lim }\left[-2g\left(x\right)\right]$

Calculation:

First, rearrange the expressions to isolate $g\left(x\right)$ . Then, substitute the second definition and solve.

  $\underset{x\to c}{\lim }\left[-2g\left(x\right)\right]=-2\underset{x\to c}{\lim }g\left(x\right)$

  $=-2\left(8\right)$

  $=-16$

Conclusion:

Hence, the limit for $\underset{x\to c}{\lim }\left[-2g\left(x\right)\right]$ is $-16$

(b)

To determine

To evaluate: the value of the given limit.

Answer to Problem 51E

Here, the limit for $\underset{x\to c}{\lim }\left[f\left(x\right)+g\left(x\right)\right]$ is 12.

Explanation of Solution

Given: $\underset{x\to c}{\lim }\left[f\left(x\right)+g\left(x\right)\right]$

Calculation:

Rearrange the expressions to isolate $f\left(x\right)$ and $g\left(x\right)$ with their respective limits. Substitute the definitions respectively and solve.

  $\underset{x\to c}{\lim }\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to c}{\lim }f\left(x\right)+\underset{x\to c}{\lim }g\left(x\right)$

  $=\left(4\right)+\left(8\right)$

  $=12$

Conclusion:

Hence, the limit for $\underset{x\to c}{\lim }\left[f\left(x\right)+g\left(x\right)\right]$ is 12.

(c)

To determine

To evaluate: the limit

Answer to Problem 51E

Here, the limit for $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is $\frac{1}{2}$ .

Explanation of Solution

Given: $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$

Calculation:

Use the law of division for limits to rearrange the expression. Substitute the definitions for each limit respectively and solve.

  $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to c}{\lim }f\left(x\right)}{\underset{x\to c}{\lim }g\left(x\right)}$

  $=\frac{\left(4\right)}{\left(8\right)}$

  $=\frac{1}{2}$

Conclusion:

Hence, the limit for $\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}$ is $\frac{1}{2}$ .

(d)

To determine

To evaluate: the value of the limit

Answer to Problem 51E

Here, the limit for $\underset{x\to c}{\lim }\sqrt{f\left(x\right)}$ is 2.

Explanation of Solution

Given: $\underset{x\to c}{\lim }\sqrt{f\left(x\right)}$

Calculation:

Use the root for limits to rearrange the expression .Substitute and solve.

  $\underset{x\to c}{\lim }\sqrt{f\left(x\right)}=\sqrt{\underset{x\to c}{\lim }f\left(x\right)}$

  $=\sqrt{\left(4\right)}$

  $=2$

Conclusion:

Hence, the limit for $\underset{x\to c}{\lim }\sqrt{f\left(x\right)}$ is 2.