# Find the horizontal and/or oblique asymptote

Find the horizontal and/or oblique asymptotes of the one system function

$$f(x) = \begin{cases} \dfrac{1-2x}{3x^2} & \text{if x < 0} \\ \dfrac{x^4+x^3+1}{x+1} & \text{if x \geq 0} \end{cases}$$

We have: $\displaystyle \lim_{x \to -\infty} f(x) = 0$, thus the horizontal asymptote is: $y = 0$.
For the oblique asymptote, we have: $x^4 + x^3 + 1 = x^3(x+1) + 1 \Rightarrow \dfrac{x^4+x^3+1}{x+1} = x^3 + \dfrac{1}{x+1}$. Thus the oblique asymptote is: $y = x^3$.