Do all functions with vertical asymptotes also have oblique asymptotes? I just started learning about asymptotes in my Advanced Functions class, and as I was taking a look at all this stuff, a question came up. Do all rational functions that have vertical asymptotes also have an oblique asymptote? Or is an oblique asymptote only formed when the degree of the numerator is 1 higher than the degree of the denominator, and so only functions with a vertical asymptote with a degree of 1 can also have an oblique asymptote?

Inbrunstlr 2022-09-29 Answered
Do all functions with vertical asymptotes also have oblique asymptotes?
I just started learning about asymptotes in my Advanced Functions class, and as I was taking a look at all this stuff, a question came up. Do all rational functions that have vertical asymptotes also have an oblique asymptote? Or is an oblique asymptote only formed when the degree of the numerator is 1 higher than the degree of the denominator, and so only functions with a vertical asymptote with a degree of 1 can also have an oblique asymptote?
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Answers (1)

Shania Delacruz
Answered 2022-09-30 Author has 7 answers
You are dealing with dividing two functions and the bottom function approaches zero while the top function approaches a non zero value.
For example
f ( x ) = 2 x + 1 ( x 5 ) ( 2 x + 3 )
where x=5 and x=−3/2 are vertical asymptotes.
Oblique asymptotes happen when your function behaves like a non-horizontal straight line as x goes to or We find slant asymptotes by dividing the top by the bottom and ignoring the remainder.
For example
f ( x ) = 2 x 2 + 1 2 x + 3
where your function behaves like
g ( x ) = x 3 / 2
which is a straight line.
A function may have both vertical and oblique asymptote but not both horizontal and slant.
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