Calculus Tutorial 2.7.019

 
Problem Statement

{2,3}
2*2
{4,5,6,7,8,9}
{4,5,6,7,8,9}
eval round(2/4,2)
eval round(-2/4,2)

Find the horizontal asymptotes

 
Step 1

Recall what it means to be a horizontal asymptote.

Question Sequence

Question 1

A horizontal line y = L is called a horizontal asymptote if = and/or = .

Incorrect.
Correct.

 
Step 2

We will first evaluate

In general, we wish to use the Quotient Law to evaluate limits of rational expressions. Recall the Quotient Law. If and exist (as a finite number) and , then we have the following equation.

Question Sequence

Question 3

To see if we can directly apply the Quotient Law to evaluate , use Limit Laws to determine the limit as x approaches ∞ of both the numerator and denominator. Recall that = for all n > 0.

Incorrect.
Correct.

 
Step 3

Since we cannot apply the Quotient Law directly to , we will divide both the numerator and the denominator by the highest power of x present in the denominator which is 1. This is equivalent to multiplying the numerator and denominator by x−1.

To make this multiplication more useful in the numerator, we will rewrite x−1 as a square root given that we are currently considering x > 0 only.

Question Sequence

Question 6

For x > 0, , where a = .

Incorrect.
Correct.

 
Step 4

We will now be able to apply the Quotient Law and other Limit Laws to the new limit from Step 3 since the limits in the numerator and denominator will be finite.

Question Sequence

Question 8

In evaluating the new limit, we will also have to use the fact that for any whole number n, we have

=

Incorrect.
Correct.

 
Step 5

Now we will evaluate

Question Sequence

Question 10

To see if we can directly apply the Quotient Law to evaluate , use Limit Laws to determine the limit as x approaches -∞ of both the numerator and denominator. Recall that = if n is even and = if n is odd.

Incorrect.
Correct.

 
Step 6

Since we cannot apply the Quotient Law directly to , we will divide both the numerator and the denominator by the highest power of x present in the denominator which is 1. This is equivalent to multiplying the numerator and denominator by x−1.

To make this multiplication more useful in the numerator, we will rewrite x−1 as a square root given that we are currently considering x > 0 only.

Question Sequence

Question 13

For x < 0, , where a = .

Incorrect.
Correct.

 
Step 7

We will now be able to apply the Quotient Law and other Limit Laws to the new limit from Step 6 since the limits in the numerator and denominator will be finite.

Question Sequence

Question 15

In evaluating the new limit, we will also have to use the fact that for any whole number n, we have

=

Incorrect.
Correct.

 
Step 8

State the horizontal asymptotes of

Question Sequence

Question 17

There are two horizontal asymptotes.

y = (smaller y-value)

y = (larger y-value)

(Round your answers to two decimal places.)

Incorrect.
Correct.