# Find the Asymptotes y=(x^2-4)/(3x-6)

Find where the expression is undefined.
The vertical asymptotes occur at areas of infinite discontinuity.
No Vertical Asymptotes
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).
Find and .
Since , there is no horizontal asymptote.
No Horizontal Asymptotes
Find the oblique asymptote using polynomial division.
Simplify the expression.
Simplify the numerator.
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Simplify terms.
Factor out of .
Factor out of .
Factor out of .
Factor out of .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
 +
Divide the highest order term in the dividend by the highest order term in divisor .
 +
Multiply the new quotient term by the divisor.
 + +
The expression needs to be subtracted from the dividend, so change all the signs in
 + –
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 + –
Pull the next terms from the original dividend down into the current dividend.
 + – +
Divide the highest order term in the dividend by the highest order term in divisor .
 + + – +
Multiply the new quotient term by the divisor.
 + + – + +
The expression needs to be subtracted from the dividend, so change all the signs in
 + + – + –
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
 + + – + –
Since the remander is , the final answer is the quotient.
The oblique asymptote is the polynomial portion of the long division result.
This is the set of all asymptotes.
No Vertical Asymptotes
No Horizontal Asymptotes
Oblique Asymptotes:
Find the Asymptotes y=(x^2-4)/(3x-6)