Chapter 13.2, Problem 4E

a.

To determine

To find: The value of the given limit. If the value does not exist, explain why.

Answer to Problem 4E

  $\underset{x\to 2}{\lim }\left[f\left(x\right)+g\left(x\right)\right]=0$

Explanation of Solution

Given information:

  

From the graph,

  $\underset{x\to 2}{\lim }f\left(x\right)=2$ ; $\underset{x\to 2}{\lim }g\left(x\right)=0$

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

Formula used:

  $\underset{x\to a}{\lim }\left[f\left(x\right)+h\left(x\right)\right]=\underset{x\to a}{\lim }f\left(x\right)+\underset{x\to a}{\lim }h\left(x\right)\text{ (Limit of sum)}$

Calculation:

  $\begin{array}{c}\underset{x\to 2}{\lim }\left[f\left(x\right)+g\left(x\right)\right]\\ =\underset{x\to 2}{\lim }f\left(x\right)+\underset{x\to 2}{\lim }g\left(x\right)\\ =2+0\\ =2\end{array}$

b.

To determine

To find: The value of the given limit. If the value does not exist, explain why.

Answer to Problem 4E

  $\underset{x\to 1}{\lim }\left[f\left(x\right)+g\left(x\right)\right]$ does not exist because , the limit $\underset{x\to 1}{\lim }g\left(x\right)$ does not exist.

Explanation of Solution

Given information:

  

From the graph,

  $\underset{x\to 1}{\lim }f\left(x\right)=1$ ; $\underset{x\to 1^{-}}{\lim }g\left(x\right)=2$ ; $\underset{x\to 1^{+}}{\lim }g\left(x\right)=1$

  $\underset{x\to 1}{\lim }\left[f\left(x\right)+g\left(x\right)\right]=?$

Formula used:

  $\underset{x\to a}{\lim }\left[f\left(x\right)+h\left(x\right)\right]=\underset{x\to a}{\lim }f\left(x\right)+\underset{x\to a}{\lim }h\left(x\right)\text{ (Limit of sum)}$

  $\underset{x\to a}{\lim }f\left(x\right)=L\text{ if and only if }\underset{x\to a^{-}}{\lim }f\left(x\right)=L\text{ and }\underset{x\to a+}{\lim }f\left(x\right)=L$

If the left-hand and right-hand limits are different, the limit does not exist.

Calculation:

  $\begin{array}{c}\underset{x\to 1}{\lim }\left[f\left(x\right)+g\left(x\right)\right]\\ =\underset{x\to 1}{\lim }f\left(x\right)+\underset{x\to 1}{\lim }g\left(x\right)\end{array}$

Now, $\underset{x\to 1^{-}}{\lim }g\left(x\right)=2$ and $\underset{x\to 1^{+}}{\lim }g\left(x\right)=1$.

Since, the left-hand and right-hand limits are different, the limit $\underset{x\to 1}{\lim }g\left(x\right)$ does not exist.

Therefore, $\underset{x\to 1}{\lim }\left[f\left(x\right)+g\left(x\right)\right]$ does not exist.

c.

To determine

To find: The value of the given limit. If the value does not exist, explain why.

Answer to Problem 4E

  $\underset{x\to 0}{\lim }\left[f\left(x\right)g\left(x\right)\right]=0$

Explanation of Solution

Given information:

  

From the graph, $\underset{x\to 0}{\lim }f\left(x\right)=0$ and the limit $\underset{x\to 0}{\lim }g\left(x\right)$ exist.

  $\underset{x\to 0}{\lim }\left[f\left(x\right)g\left(x\right)\right]=?$

Formula used:

  $\underset{x\to 0}{\lim }\left[f\left(x\right)g\left(x\right)\right]=\underset{x\to 0}{\lim }f\left(x\right).\underset{x\to 0}{\lim }g\left(x\right)\text{ Limit of Product}$

Calculation:

  $\begin{array}{c}\underset{x\to 0}{\lim }\left[f\left(x\right)g\left(x\right)\right]\\ =\underset{x\to 0}{\lim }f\left(x\right).\underset{x\to 0}{\lim }g\left(x\right)\\ =0\times \underset{x\to 0}{\lim }g\left(x\right)\\ =0\end{array}$

d.

To determine

To find: The value of the given limit. If the value does not exist, explain why.

Answer to Problem 4E

Given limit does not exist.

Explanation of Solution

Given information:

  

From the graph, $\underset{x\to -1}{\lim }f\left(x\right)=-1$ ; $\underset{x\to -1}{\lim }g\left(x\right)=0$

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

Formula used:

  $\underset{x\to a}{\lim }\frac{f\left(x\right)}{g\left(x\right)}=\frac{\underset{x\to a}{\lim }f\left(x\right)}{\underset{x\to a}{\lim }g\left(x\right)}\text{ if }\underset{x\to a}{\lim }g\left(x\right)\ne 0$

Calculation:

  $\begin{array}{l}\underset{x\to -1}{\lim }\frac{f\left(x\right)}{g\left(x\right)}\\ =\frac{\underset{x\to -1}{\lim }f\left(x\right)}{\underset{x\to -1}{\lim }g\left(x\right)}\end{array}$

Now, $\underset{x\to -1}{\lim }g\left(x\right)=0$

Therefore, the denominator of the limit is zero. So, the limit does not exist.

e.

To determine

To find: The value of the given limit. If the value does not exist, explain why.

Answer to Problem 4E

  $\underset{x\to 2}{\lim }{x}^{3}f\left(x\right)=16$

Explanation of Solution

Given information:

  

From the graph, $\underset{x\to 2}{\lim }f\left(x\right)=2$.

  $\underset{x\to 2}{\lim }{x}^{3}f\left(x\right)=?$

Formula used:

  $\underset{x\to 0}{\lim }\left[f\left(x\right)g\left(x\right)\right]=\underset{x\to 0}{\lim }f\left(x\right).\underset{x\to 0}{\lim }g\left(x\right)\text{ Limit of Product}$

  $\underset{x\to a}{\lim }{x}^n=a^n\text{ Where }n\text{ is a positive integer}$

Calculation:

  $\begin{array}{c}\underset{x\to 2}{\lim }{x}^{3}f\left(x\right)\\ =\underset{x\to 2}{\lim }{x}^3.\underset{x\to 2}{\lim }f\left(x\right)\\ =2^3\times 2\\ =8\times 2\\ =16\end{array}$

f.

To determine

To find: The value of the given limit. If the value does not exist, explain why.

Answer to Problem 4E

  $\underset{x\to 1}{\lim }\sqrt{3+f\left(x\right)}=2$

Explanation of Solution

Given information:

  

From the graph, $\underset{x\to 1}{\lim }f\left(x\right)=1$.

  $\underset{x\to 1}{\lim }\sqrt{3+f\left(x\right)}=?$

Formula used:

  $\underset{x\to 1}{\lim }\sqrt[n]{f\left(x\right)}=\sqrt[n]{\underset{x\to 1}{\lim }f\left(x\right)}\text{ Where }n\text{ is a positive interger}$

  $\underset{x\to a}{\lim }\left[f\left(x\right)+h\left(x\right)\right]=\underset{x\to a}{\lim }f\left(x\right)+\underset{x\to a}{\lim }h\left(x\right)\text{ (Limit of sum)}$

  $\underset{x\to a}{\lim }c=c\text{ Where }c\text{ is a constant}$

Calculation:

  $\begin{array}{c}\underset{x\to 1}{\lim }\sqrt{3+f\left(x\right)}\\ =\sqrt{\underset{x\to 1}{\lim }\left(3+f\left(x\right)\right)}\\ =\sqrt{\underset{x\to 1}{\lim }3+\underset{x\to 1}{\lim }f\left(x\right)}\\ =\sqrt{3+1}\\ =\sqrt{4}\\ =2\end{array}$