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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$\begingroup$You are dealing with dividing two functions and the bottom function approaches zero while the top function approaches a non zero value.
For example $$ f(x) = \frac {2x+1}{(x-5)(2x+3)}$$ 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 $\infty$ or $-\infty$We find slant asymptotes by dividing the top by the bottom and ignoring the remainder.
For example $$ f(x) = \frac {2x^2+1}{2x+3}$$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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