Chapter 2.1, Problem 56E

(a)

To determine

To find: The value of $\underset{x\to b}{\lim }\left(f\left(x\right)+g\left(x\right)\right)$ .

Answer to Problem 56E

The value of $\underset{x\to b}{\lim }\left(f\left(x\right)+g\left(x\right)\right)$ is $4$ .

Explanation of Solution

Given information: The values $\underset{x\to b}{\lim }f\left(x\right)=7$ and $\underset{x\to b}{\lim }g\left(x\right)=-3$ .

Calculation:

Use the sum property for limit.

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

Simplify the given limit.

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

Substitute $7$ for $\underset{x\to b}{\lim }f\left(x\right)$ and $-3$ for $\underset{x\to b}{\lim }g\left(x\right)$ in the above expression.

  $\begin{array}{c}\underset{x\to b}{\lim }\left(f\left(x\right)+g\left(x\right)\right)=7+\left(-3\right)\\ =7-3\\ =4\end{array}$

Therefore, the value of $\underset{x\to b}{\lim }\left(f\left(x\right)+g\left(x\right)\right)$ is $4$ .

(b)

To determine

To find: The value of $\underset{x\to b}{\lim }\left(f\left(x\right)\cdot g\left(x\right)\right)$ .

Answer to Problem 56E

The value of $\underset{x\to b}{\lim }\left(f\left(x\right)\cdot g\left(x\right)\right)$ is $-21$ .

Explanation of Solution

Given information: The values $\underset{x\to b}{\lim }f\left(x\right)=7$ and $\underset{x\to b}{\lim }g\left(x\right)=-3$ .

Calculation:

Use the product rule for limit.

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

Simplify the given limit.

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

Substitute $7$ for $\underset{x\to b}{\lim }f\left(x\right)$ and $-3$ for $\underset{x\to b}{\lim }g\left(x\right)$ in the above expression.

  $\begin{array}{c}\underset{x\to b}{\lim }\left(f\left(x\right)\cdot g\left(x\right)\right)=7\times \left(-3\right)\\ =-21\end{array}$

Therefore, the value of $\underset{x\to b}{\lim }\left(f\left(x\right)\cdot g\left(x\right)\right)$ is $-21$ .

(c)

To determine

To find: The value of $\underset{x\to b}{\lim }\left(4g\left(x\right)\right)$ .

Answer to Problem 56E

The value of $\underset{x\to b}{\lim }\left(4g\left(x\right)\right)$ is $-12$ .

Explanation of Solution

Given information: The values $\underset{x\to b}{\lim }f\left(x\right)=7$ and $\underset{x\to b}{\lim }g\left(x\right)=-3$ .

Calculation:

Use the constant multiple rule for limit.

  $\underset{x\to a}{\lim }\left(kf\left(x\right)\right)=k\underset{x\to a}{\lim }\left(f\left(x\right)\right)$

Simplify the given limit.

  $\underset{x\to b}{\lim }\left(4g\left(x\right)\right)=4\underset{x\to b}{\lim }\left(g\left(x\right)\right)$

Substitute $-3$ for $\underset{x\to b}{\lim }\left(g\left(x\right)\right)$ in the above expression.

  $\begin{array}{c}\underset{x\to b}{\lim }\left(4g\left(x\right)\right)=4\times \left(-3\right)\\ =-12\end{array}$

Therefore, the value of $\underset{x\to b}{\lim }\left(4g\left(x\right)\right)$ is $-12$ .

(d)

To determine

To find: The value of $\underset{x\to b}{\lim }\left(\frac{f\left(x\right)}{g\left(x\right)}\right)$ .

Answer to Problem 56E

The value of $\underset{x\to b}{\lim }\left(\frac{f\left(x\right)}{g\left(x\right)}\right)$ is $-\frac{7}{3}$ .

Explanation of Solution

Given information: The values $\underset{x\to b}{\lim }f\left(x\right)=7$ and $\underset{x\to b}{\lim }g\left(x\right)=-3$ .

Calculation:

Use the quotient rule for limit.

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

Simplify the given limit.

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

Substitute $7$ for $\underset{x\to b}{\lim }\left(f\left(x\right)\right)$ and $-3$ for $\underset{x\to b}{\lim }\left(g\left(x\right)\right)$ in the above expression.

  $\begin{array}{c}\underset{x\to b}{\lim }\left(\frac{f\left(x\right)}{g\left(x\right)}\right)=\frac{7}{-3}\\ =-\frac{7}{3}\end{array}$

Therefore, the value of $\underset{x\to b}{\lim }\left(\frac{f\left(x\right)}{g\left(x\right)}\right)$ is $-\frac{7}{3}$ .