On predicting whether a real valued function has a non linear asymptote or not.

Question

Suppose I have two functions.

1. $$y= \frac{x^3+2x +9}{\sqrt{4x^2+3x+2}}$$ which has an non linear asymptote of $y=\dfrac{x^2}2-\dfrac{3x}{16}+\dfrac{251}{256}$.

2. $$y = \frac{x}{(x^4 + 1)^{1/4}}$$ which has a linear asymptote of $y=1$.

How can I predict which function may have a non linear asymptote?

Answer 1

HINT: $$f(x)=\frac{x}{\sqrt[4]{x^4+1}}=\frac{x}{|x|\sqrt[4]{1+\frac{1}{x^4}}}$$