The graph $y = f(x)$ has a slant asymptote along the line $y = mx + b$ (with $m \not= 0$) if $$ \displaystyle\lim_{x \rightarrow \infty} |f(x) - (mx+b)| = 0. $$ Describe algebraically the conditions for a rational function $\dfrac{f(x)}{g(x)}$ to have a slant asymptote, where $f$ and $g$ are polynomials.
I'm not sure how to start and do this problem, but any help is appreciated!
$\endgroup$ 31 Answer
$\begingroup$If the degree of $f(x)$ is exactly one more than the degree of $g(x)$ then
$$\frac{f(x)}{g(x)}=ax+b+\frac{c}{g(x)}$$
where $c$ is a constant and where
$$ y=ax+b $$
is an asymptote of $y=\frac{f(x)}{g(x)}$
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