Subtract from both sides of the equation.

Divide each term by and simplify.

Divide each term in by .

Cancel the common factor of .

Cancel the common factor.

Divide by .

Move the negative in front of the fraction.

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.

Combine.

Combine the numerators over the common denominator.

Simplify.

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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Divide the highest order term in the dividend by the highest order term in divisor .

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Multiply the new quotient term by the divisor.

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The expression needs to be subtracted from the dividend, so change all the signs in

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After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

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Pull the next terms from the original dividend down into the current dividend.

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Divide the highest order term in the dividend by the highest order term in divisor .

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Multiply the new quotient term by the divisor.

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The expression needs to be subtracted from the dividend, so change all the signs in

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After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

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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:

No Vertical Asymptotes

No Horizontal Asymptotes

Oblique Asymptotes:

Graph Ax+By=C