Calculus Examples

Find the Asymptotes y=(x^2-x-2)/(x-2)
Step 1
Find where the expression is 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
Find the oblique asymptote using polynomial division.
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Step 6.1
Simplify the expression.
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Step 6.1.1
Factor using the AC method.
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Step 6.1.1.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 6.1.1.2
Write the factored form using these integers.
Step 6.1.2
Cancel the common factor of .
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Step 6.1.2.1
Cancel the common factor.
Step 6.1.2.2
Divide by .
Step 6.2
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