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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Shania Delacruz · Accepted Answer

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      /    2which is a straight line.A function may have both vertical and oblique asymptote but not both horizontal and slant.

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