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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!

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    If the degree f=degree g+1. – 2017-02-05
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    Hint: the slope would have to be $m = \lim_{x \to \infty} \frac{f(x)}{x\,g(x)}\,$. Now look at the degrees of $f,g$ and see when that limit can exist and be non-zero. – 2017-02-05
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    Essentially, the degree of $f(x)$ must be $1$ more than the degree of $g(x)$ so that the quotient is of degree $1$. – 2017-02-05

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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 $y=\frac{f(x)}{g(x)}$