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Precalculus Examples
Step 1
Find where the expression is undefined.
The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.
Step 2
The vertical asymptotes occur at areas of infinite discontinuity.
No Vertical Asymptotes
Step 3
Consider the rational function where is the degree of the numerator and is the degree of the denominator.
1. If , then the x-axis, , is the horizontal asymptote.
2. If , then the horizontal asymptote is the line .
3. If , then there is no horizontal asymptote (there is an oblique asymptote).
Step 4
Find and .
Step 5
Since , there is no horizontal asymptote.
No Horizontal Asymptotes
Step 6
Step 6.1
Simplify the numerator.
Step 6.1.1
Rewrite as .
Step 6.1.2
Since both terms are perfect cubes, factor using the sum of cubes formula, where and .
Step 6.1.3
Simplify.
Step 6.1.3.1
Multiply by .
Step 6.1.3.2
One to any power is one.
Step 6.2
Expand .
Step 6.2.1
Apply the distributive property.
Step 6.2.2
Apply the distributive property.
Step 6.2.3
Apply the distributive property.
Step 6.2.4
Apply the distributive property.
Step 6.2.5
Apply the distributive property.
Step 6.2.6
Remove parentheses.
Step 6.2.7
Reorder and .
Step 6.2.8
Reorder and .
Step 6.2.9
Remove parentheses.
Step 6.2.10
Multiply by .
Step 6.2.11
Raise to the power of .
Step 6.2.12
Use the power rule to combine exponents.
Step 6.2.13
Add and .
Step 6.2.14
Factor out negative.
Step 6.2.15
Raise to the power of .
Step 6.2.16
Raise to the power of .
Step 6.2.17
Use the power rule to combine exponents.
Step 6.2.18
Add and .
Step 6.2.19
Multiply by .
Step 6.2.20
Multiply by .
Step 6.2.21
Multiply by .
Step 6.2.22
Multiply by .
Step 6.2.23
Move .
Step 6.2.24
Move .
Step 6.2.25
Subtract from .
Step 6.2.26
Add and .
Step 6.2.27
Subtract from .
Step 6.2.28
Add and .
Step 6.3
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
+ | + | + | + | + |
Step 6.4
Divide the highest order term in the dividend by the highest order term in divisor .
+ | + | + | + | + |
Step 6.5
Multiply the new quotient term by the divisor.
+ | + | + | + | + | |||||||||
+ | + | + |
Step 6.6
The expression needs to be subtracted from the dividend, so change all the signs in
+ | + | + | + | + | |||||||||
- | - | - |
Step 6.7
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
+ | + | + | + | + | |||||||||
- | - | - | |||||||||||
- |
Step 6.8
Pull the next term from the original dividend down into the current dividend.
+ | + | + | + | + | |||||||||
- | - | - | |||||||||||
- | + |
Step 6.9
The final answer is the quotient plus the remainder over the divisor.
Step 6.10
The oblique asymptote is the polynomial portion of the long division result.
Step 7
This is the set of all asymptotes.
No Vertical Asymptotes
No Horizontal Asymptotes
Oblique Asymptotes:
Step 8