The question here elaborates on the common method to find asymptotes—divide and the quotient's your answer. I understand this, and also why it works. However, my book has a rather different definition:
If the limits $$ \lim_{x\to∞}\frac{f(x)}{x} = k,\ \lim_{x\to∞}\big(f(x) -kx\big) = b $$ exist, then, the straight line $y = kx+b$ is an inclined (right) asymptote
and likewise for the inclined left asymptote as $x\to-∞$. Why is this correct, and where does this come from? Google searches only yield the long division approach, and nothing about this one.