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 $\frac{-7}{x+3}$?