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QUESTION

How do you find vertical, horizontal and oblique asymptotes for ##(-2x^2-6x+3)/(3x+9)##?

Vertical asymptote: ##x=-3## Horizontal asymptote: none Oblique asymptote: ##y=-2x##

How to find vertical asymptote:

Set denominator to equal zero and solve for ##x##

##3x+9## ##rarr## ##3x=-9## ##rarr## ##x=-9/3## ##rarr## ##x=-3##

How to find horizontal asymptote:

The degree in the numerator is greater than the degree of the denominator, therefore there is no horizontal asymptote.

How to find oblique asymptote:

The degree in the numerator is one more than the degree in the denominator, therefore there is a slant/oblique asymptote present. To find the slant asymptote, you must make sure that the numerator is in quadratic form. In this case, since the numerator is already in quadratic form, we leave it as it is. From here, we then perform synthesis division to find the slant asymptote.

We don't have to worry about the remainder column in this situation. This makes the slant asymptote: ##y=-2x## We also know that there is a slant asymptote because there is also a vertical asymptote present.