Chapter 11.1, Problem 52E

(a)

To determine

To evaluate: the limit

Answer to Problem 52E

Here, the limit of $\underset{x\to c}{\lim }{\left[f\left(x\right)+g\left(x\right)\right]}^2$ is 4 .

Explanation of Solution

Given:

  $\underset{x\to c}{\lim }f\left(x\right)=3$

  $\underset{x\to c}{\lim }g\left(x\right)=-1$

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

Calculation:

Use the power law of limits and the addition lawof limits to rearrange the equation. Then, substitute the definitions for each limit and simplify.

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

  $={\left(\underset{x\to c}{\lim }f\left(x\right)+\underset{x\to c}{\lim }g\left(x\right)\right)}^2$

  $={\left[\left(3\right)+\left(-1\right)\right]}^2$

  $=2^2$

  $=4$

Conclusion:

Hence, the limit of $\underset{x\to c}{\lim }{\left[f\left(x\right)+g\left(x\right)\right]}^2$ is 4 .

(b)

To determine

To evaluate: the limit

Answer to Problem 52E

Here, the limit of $\underset{x\to c}{\lim }\left[6f\left(x\right)g\left(x\right)\right]$ is -18.

Explanation of Solution

Given: $\underset{x\to c}{\lim }\left[6f\left(x\right)g\left(x\right)\right]$

Calculation:

Use the law of multiplication for limits twice to rearrange the expression. Then, substitute the definitions for each limit and simplify.

  $\underset{x\to c}{\lim }\left[6f\left(x\right)g\left(x\right)\right]=6\underset{x\to c}{\lim }\left[f\left(x\right)g\left(x\right)\right]$

  $=6\underset{x\to c}{\lim }f\left(x\right)\cdot \underset{x\to c}{\lim }g\left(x\right)$

  $=6\left(3\right)\left(-1\right)$

  $=-18$

Conclusion:

Hence, the limit of $\underset{x\to c}{\lim }\left[6f\left(x\right)g\left(x\right)\right]$ is -18.

(c)

To determine

To evaluate: the limit

Answer to Problem 52E

Here, the limit of $\underset{x\to c}{\lim }\frac{5g\left(x\right)}{4f\left(x\right)}$ is $-\frac{15}{4}$ .

Explanation of Solution

Given: $\underset{x\to c}{\lim }\frac{5g\left(x\right)}{4f\left(x\right)}$

Calculation:

Part C

Use the law of divisions for limits and then the law of multiplications for limits to rearrange the expression. Substitute the definitions and simplify.

  $\underset{x\to c}{\lim }\frac{5f\left(x\right)}{4g\left(x\right)}=\frac{\underset{x\to c}{\lim }\left[5f\left(x\right)\right]}{\underset{x\to c}{\lim }\left[4g\left(x\right)\right]}$

  $=\frac{5\underset{x\to c}{\lim }f\left(x\right)}{4\underset{x\to c}{\lim }g\left(x\right)}$

  $=\frac{5\left(3\right)}{4\left(-1\right)}$

  $=-\frac{15}{4}$

Conclusion:

Hence, the limit of $\underset{x\to c}{\lim }\frac{5g\left(x\right)}{4f\left(x\right)}$ is $-\frac{15}{4}$ .

(d)

To determine

To evaluate: the limit

Answer to Problem 52E

Here, the limit of $\underset{x\to c}{\lim }\frac{1}{\sqrt{f\left(x\right)}}$ is $\frac{\sqrt{3}}{3}$ .

Explanation of Solution

Given: $\underset{x\to c}{\lim }\frac{1}{\sqrt{f\left(x\right)}}$

Calculation:

Use the division law and the root law for limits to rearrange the expression. Next, substitute the definitions and simplify.

  $\underset{x\to c}{\lim }\frac{1}{\sqrt{f\left(x\right)}}=\frac{\underset{x\to c}{\lim }1}{\underset{x\to c}{\lim }\sqrt{f\left(x\right)}}$

  $=\frac{1}{\sqrt{\underset{x\to c}{\lim }f\left(x\right)}}$

  $=\frac{1}{\sqrt{3}}$

  $=\frac{1}{\sqrt{3}}\cdot \frac{\sqrt{3}}{\sqrt{3}}$

  $=\frac{\sqrt{3}}{3}$

Conclusion:

Hence,the limit of $\underset{x\to c}{\lim }\frac{1}{\sqrt{f\left(x\right)}}$ is $\frac{\sqrt{3}}{3}$ .