Chapter 1. Calculus Tutorial 2.7.019

1.1 Problem Statement

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

Find the horizontal asymptotes

1.2 Step 1

Recall what it means to be a horizontal asymptote.

Question Sequence

Question 1.1

A horizontal line y = L is called a horizontal asymptote if = M1jvgaUwpgE= and/or = M1jvgaUwpgE=.

Incorrect.
Correct.

Question 1.2

Therefore, to find the horizontal asymptotes of the graph of we must evaluate both and . If these limits are ucK/0066/j/hcQZKHtwSKNM+GL4=, then we will have horizontal asymptotes.

Incorrect.
Correct.

1.3 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 1.3

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 = SfbBPKJrgP2mt2D6l8so4paymHo= for all n > 0.

Incorrect.
Correct.

Question 1.4

= 3LecBr2w/JA=

= 3LecBr2w/JA=

(Type "inf" for ∞ and "-inf" for -∞)

Incorrect.
Correct.

Question 1.5

Since both of these limits sGe9BG4ptbLdvBlRE8fTlQ== exist as finite numbers, we 98RZuska2DoKNkUANZwGUQ== directly apply the Quotient Law.

Incorrect.
Correct.

1.4 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 1.6

For x > 0, , where a = FmAhxBkQqN0=.

Incorrect.
Correct.

Question 1.7

, where a = FmAhxBkQqN0=.

, where y = XvVM00l89Is= and w = SFgqQUkJGdg=.

Incorrect.
Correct.

1.5 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 1.8

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

= 1Wh3cvJ2xF4=

Incorrect.
Correct.

Question 1.9

Use Limit Laws to evaluate .

= dQd7hw/ACY+iL07C

Therefore, we have y = dQd7hw/ACY+iL07C as the horizontal asymptote to f(x) by definition.

(Round your answer to two decimal places.)

Incorrect.
Correct.

1.6 Step 5

Now we will evaluate

Question Sequence

Question 1.10

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 = jsxGUCDJgxln/k+r839zcA== if n is even and = HC/PNLn1U962ptkIzLahvw== if n is odd.

Incorrect.
Correct.

Question 1.11

= 3LecBr2w/JA=

= ugHNTI1hRxM=

(Type "inf" for ∞ and "-inf" for -∞)

Incorrect.
Correct.

Question 1.12

Since both of these limits tlT+43pgkbTptNAbfAd2Sw== exist as finite numbers, we 98RZuska2DoKNkUANZwGUQ== directly apply the Quotient Law.

Correct.
Incorrect.

1.7 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 1.13

For x < 0, , where a = FmAhxBkQqN0=.

Incorrect.
Correct.

Question 1.14

, where a = FmAhxBkQqN0=.

, where y = XvVM00l89Is= and w = SFgqQUkJGdg=.

Incorrect.
Correct.

1.8 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 1.15

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

= 1Wh3cvJ2xF4=

Incorrect.
Correct.

Question 1.16

Use Limit Laws to evaluate .

= Qbq9sudn0aydEORe

Therefore, we have y = Qbq9sudn0aydEORe as the horizontal asymptote to f(x) by definition.

(Round your answer to two decimal places.)

Incorrect.
Correct.

1.9 Step 8

State the horizontal asymptotes of

Question Sequence

Question 1.17

There are two horizontal asymptotes.

y = Qbq9sudn0aydEORe (smaller y-value)

y = dQd7hw/ACY+iL07C (larger y-value)

(Round your answers to two decimal places.)

Incorrect.
Correct.