Chapter 11, Problem 9RE

(a)

To determine

To evaluate: the function

Answer to Problem 9RE

The solution is 8.

Explanation of Solution

Given:

  $\begin{array}{l}\underset{x\to c}{\lim }f\left(x\right)=2\\ \underset{x\to c}{\lim }g\left(x\right)=5\end{array}$

Calculation:

Part A

Rearrange the expression and use the defined limits to solve the problem.

  $\begin{array}{l}\underset{x\to c}{\lim }{\left[f\left(x\right)\right]}^3\\ ={\left[\underset{x\to c}{\lim }f\left(x\right)\right]}^3\\ ={\left[\left(2\right)\right]}^3\\ =8\end{array}$

Conclusion:

Therefore,the solution is 8.

(b)

To determine

To evaluate: the function

Answer to Problem 9RE

The solution is 1.

Explanation of Solution

Given:

  $\begin{array}{l}\underset{x\to c}{\lim }f\left(x\right)=2\\ \underset{x\to c}{\lim }g\left(x\right)=5\end{array}$

Calculation:

Part B:

Rearrange the expression and use the defined limits to solve the problem.

  $\begin{array}{l}\underset{x\to c}{\lim }\left[3f\left(x\right)-g\left(x\right)\right]\\ =\underset{x\to c}{\lim }\left[3f\left(x\right)\right]-\underset{x\to c}{\lim }g\left(x\right)\\ =3\underset{x\to c}{\lim }f\left(x\right)-\underset{x\to c}{\lim }g\left(x\right)\\ =3\left(2\right)-\left(5\right)\\ =6-5\\ =1\end{array}$

Conclusion:

Therefore,the solution is 1.

(c)

To determine

To evaluate: the function

Answer to Problem 9RE

The solution is10.

Explanation of Solution

Given:

  $\begin{array}{l}\underset{x\to c}{\lim }f\left(x\right)=2\\ \underset{x\to c}{\lim }g\left(x\right)=5\end{array}$

Calculation:

Part C:

Rearrange the expression and use the defined limits to solve the problem.

  $\begin{array}{l}\underset{x\to c}{\lim }\left[f\left(x\right)g\left(x\right)\right]\\ =\underset{x\to c}{\lim }f\left(x\right)\cdot \underset{x\to c}{\lim }g\left(x\right)\\ =\left(2\right)-\left(5\right)\\ =10\end{array}$

Conclusion:

Therefore,the solution is10.

(d)

To determine

To evaluate: the function

Answer to Problem 9RE

The solution is $\frac{2}{5}$ .

Explanation of Solution

Given:

  $\begin{array}{l}\underset{x\to c}{\lim }f\left(x\right)=2\\ \underset{x\to c}{\lim }g\left(x\right)=5\end{array}$

Calculation:

Part C:

Rearrange the expression and use the defined limits to solve the problem.

  $\begin{array}{l}\underset{x\to c}{\lim }\frac{f\left(x\right)}{g\left(x\right)}\\ =\frac{\underset{x\to c}{\lim }f\left(x\right)}{\underset{x\to c}{\lim }g\left(x\right)}\\ =\frac{\left(2\right)}{\left(5\right)}\\ =\frac{2}{5}\end{array}$

Conclusion:

Therefore,the solution is $\frac{2}{5}$ .