# How to know if the derivative of a function has an asymptote

This is my function $$f(x)$$

What is the methodology for me determining if the sketch of the derivative has a horizontal asymptote? basically i want to be able to justify that it has a horizontal asymptote and not just because it looks like it does

• Show that $f(x)\to 0$ as $x\to\pm\infty$, and show the sequence if functions is non zero. – Pixel Jun 2 at 12:36

By looking at your picture (with no values, or scale) it seems that for large positive $$x$$ and for large negative $$x$$ the slope is $$0$$. Thus, the value of the derivate for large positive $$x$$ and large negative $$x$$ is $$0$$. This suggest that the derivative of $$f$$, i.e $$f'$$ has the horizontal asymptot $$y=0$$.
Without additional info its hard to say much more. And it also could be false, since we don't know how much of $$f$$ is shown in the picture.
• Yes. But if the function is only shown for lets say $x$ from $-0.1$ to $0.1$ then its hard to talk about large $x$ right? You can have a idea, but not much to back it up. @user130306 – Olba12 Jun 2 at 9:10
In general, given a function $$f(x)$$, you can evaluate $$\lim_{x \to \infty} f(x) \ \ \textrm{and} \ \ \lim_{x \to -\infty} f(x)$$ to get the horizontal asymptotes. There are at most two horizontal asymptotes since there are only two directions.