- #1

- 571

- 162

Playing with some numerical simulations, I plotted this in Wolfram Cloud / Mathematica:

##x^3-\frac{x^5}{x^2+2}##

I had naively expected it to approach ##x^3−x^3=0##, but that isn't the case. It approaches 2x.

I can now vaguely understand that the two terms need not cancel at infinity, but I'd like to get a better handle on this.

[1] How to break this down intuitively and estimate the qualitative nature of the asymptote "by inspection"?

[2] How can we obtain the asymptote ##2x## theoretically?

[3] Can we find the equation of the asymptote using Mathematica etc?

##x^3-\frac{x^5}{x^2+2}##

I had naively expected it to approach ##x^3−x^3=0##, but that isn't the case. It approaches 2x.

I can now vaguely understand that the two terms need not cancel at infinity, but I'd like to get a better handle on this.

[1] How to break this down intuitively and estimate the qualitative nature of the asymptote "by inspection"?

[2] How can we obtain the asymptote ##2x## theoretically?

[3] Can we find the equation of the asymptote using Mathematica etc?

**[Moderator's note: Moved from a technical forum and thus no template.]**
Last edited by a moderator: