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I understand how and when to calculate slant asymptotes of rational functions with numerators with one degree higher than the denominator, but I am confused as to why we can disregard the remainder when calculating the slant asymptote of the rational function. I have looked online but all of the websites I’ve looked at just say to discard the remainder and don’t explain why.

Say, for example that you were calculating the slant asymptote of the function $f(x) = x^2+3x-\frac{7}{x+3}$ and you calculated the slant asymptote to be $f(x) = x + \frac{-7}{x+3}$. Why is it that when graphing the slant asymptote, you can just graph the slant asymptote as $f(x) = x$ and disregard the $\frac{-7}{x+3}$?

Bruno B
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Declan H
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  • Hi and welcome to MSE! You should try typing math using MathJax (pretty much LaTeX if you're familiar with that). You can right-click existing math expressions and choose Show Math As -> TeX Commands to see how other users have written theirs, and there should be available guides on the net and in the Meta side of MSE. Otherwise, have a nice day! – Bruno B Feb 07 '24 at 15:57
  • As for your question, I'd say it's because it goes to $0$ at infinity, hence making it not impactful. That's my two cents, which should be reasonable. – Bruno B Feb 07 '24 at 16:00

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