Precalculus Examples

Find the Asymptotes f(x)=(x^2+3x+1)/(x-2)
Step 1
Find where the expression is undefined.
Step 2
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 3
Find and .
Step 4
Since , there is no horizontal asymptote.
No Horizontal Asymptotes
Step 5
Find the oblique asymptote using polynomial division.
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Step 5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
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Step 5.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 5.3
Multiply the new quotient term by the divisor.
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Step 5.4
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 5.6
Pull the next terms from the original dividend down into the current dividend.
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Step 5.7
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 5.8
Multiply the new quotient term by the divisor.
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Step 5.9
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 5.11
The final answer is the quotient plus the remainder over the divisor.
Step 5.12
The oblique asymptote is the polynomial portion of the long division result.
Step 6
This is the set of all asymptotes.
Vertical Asymptotes:
No Horizontal Asymptotes
Oblique Asymptotes:
Step 7