Chapter 2.4, Problem 1E

a.

To determine

To express : the given limits as $\infty ,-\infty ,orDNE\left(DoesNotExist\right)$ .

Answer to Problem 1E

  $\underset{x\to 4^{-}}{\lim }f\left(x\right)=\boxed{-\infty }$

Explanation of Solution

Given information:. $f\left(x\right)=\frac{5}{x-4};a=4$

Concept Involved:

Limit means intended height of the function.

  $\underset{x\to a^{-}}{\lim }f\left(x\right)$ means left hand limit.

Left hand limit indicates the height of function as x approaches $a$ from the left hand side.

We need to check the behavior of the function as x approaches $a$ from the left hand side.

In the same way, $\underset{x\to a^{+}}{\lim }f\left(x\right)$ means right hand limit.

Right hand limit indicates the height of function as x approaches $a$ from the right hand side.

We need to check the behavior of the function as x approaches $a$ from the right hand side.

If $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)=L$ , then we can conclude that $\underset{x\to a}{\lim }f\left(x\right)=L$

If $\underset{x\to a^{-}}{\lim }f\left(x\right)\ne \underset{x\to a^{+}}{\lim }f\left(x\right)$ , then we can conclude that $\underset{x\to a}{\lim }f\left(x\right)=DNE$

Calculation:

Picking the x values from the left of   $a=4$	$f\left(x\right)=\frac{5}{x-4}$
$x=3$	$\frac{5}{3-4}=-5$
$x=3.9$	$\frac{5}{3.9-4}=-50$
$x=3.99999$	$\frac{5}{3.99999-4}=-500000$
$x=3.99999999999$	$\frac{5}{3.9999999999-4}=-50000000000$

Graph

  

Interpretation:

From the table and the end behavior of graph we can see that as x is getting closer to 4 from left, the f(x) is getting closer to $-\infty$ , So $\underset{x\to 4^{-}}{\lim }\frac{5}{x-4}=-\infty$

b.

To determine

To express : the given limits as $\infty ,-\infty ,orDNE\left(DoesNotExist\right)$ .

Answer to Problem 1E

  $\underset{x\to 4^{+}}{\lim }f\left(x\right)=\boxed{\infty }$

Explanation of Solution

Given information:. $f\left(x\right)=\frac{5}{x-4};a=4$

Concept Involved:

Limit means intended height of the function.

  $\underset{x\to a^{-}}{\lim }f\left(x\right)$ means left hand limit.

Left hand limit indicates the height of function as x approaches $a$ from the left hand side.

We need to check the behavior of the function as x approaches $a$ from the left hand side.

In the same way, $\underset{x\to a^{+}}{\lim }f\left(x\right)$ means right hand limit.

Right hand limit indicates the height of function as x approaches $a$ from the right hand side.

We need to check the behavior of the function as x approaches $a$ from the right hand side.

If $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)=L$ , then we can conclude that $\underset{x\to a}{\lim }f\left(x\right)=L$

If $\underset{x\to a^{-}}{\lim }f\left(x\right)\ne \underset{x\to a^{+}}{\lim }f\left(x\right)$ , then we can conclude that $\underset{x\to a}{\lim }f\left(x\right)=DNE$

Calculation:

Picking the x values from the left of   $a=4$	$f\left(x\right)=\frac{5}{x-4}$
$x=5$	$\frac{5}{5-4}=5$
$x=4.1$	$\frac{5}{4.1-4}=50$
$x=4.0000001$	$\frac{5}{4.0000001-4}=50000000$
$x=4.0000000001$	$\frac{5}{4.0000000001-4}=50000000000$

Graph

  

Interpretation:

From the table and the end behavior of graph we can see that as x is getting closer to 4 from right, the f(x) is getting closer to $\infty$ , So $\underset{x\to 4^{+}}{\lim }\frac{5}{x-4}=\infty$

c.

To determine

To express : the given limits as $\infty ,-\infty ,orDNE\left(DoesNotExist\right)$ .

Answer to Problem 1E

  $\underset{x\to 4}{\lim }f\left(x\right)=\boxed{DNE}\left(DoesNotExist\right)$

Explanation of Solution

Given information:. $f\left(x\right)=\frac{5}{x-4};a=4$

Concept Involved:

Limit means intended height of the function.

  $\underset{x\to a^{-}}{\lim }f\left(x\right)$ means left hand limit.

Left hand limit indicates the height of function as x approaches $a$ from the left hand side.

We need to check the behavior of the function as x approaches $a$ from the left hand side.

In the same way, $\underset{x\to a^{+}}{\lim }f\left(x\right)$ means right hand limit.

Right hand limit indicates the height of function as x approaches $a$ from the right hand side.

We need to check the behavior of the function as x approaches $a$ from the right hand side.

If $\underset{x\to a^{-}}{\lim }f\left(x\right)=\underset{x\to a^{+}}{\lim }f\left(x\right)=L$ , then we can conclude that $\underset{x\to a}{\lim }f\left(x\right)=L$

If $\underset{x\to a^{-}}{\lim }f\left(x\right)\ne \underset{x\to a^{+}}{\lim }f\left(x\right)$ , then we can conclude that $\underset{x\to a}{\lim }f\left(x\right)=DNE$

Calculation:

We found

  $\begin{array}{l}\underset{x\to 4^{-}}{\lim }\frac{5}{x-4}=-\infty \text{}\\ \underset{x\to 4^{+}}{\lim }\frac{5}{x-4}=\infty \end{array}$

Graph

  

Interpretation:

Since the left hand limit is not equal to right hand limit as x approaches 4, for this given function

  $f\left(x\right)=\frac{5}{x-4}$ ,we can conclude that $\begin{array}{l}\text{}\\ \underset{x\to 4}{\lim }\frac{5}{4-x}=DNE\left(DoesNotExist\right)\end{array}$