65F. Terminal behaviour - JulTob/Mathematics GitHub Wiki

We may be interested, for many reasons, as to how a function behaves on an unlimited increase or decrease of $x$ as it goes away from $0$

  • $x⟶∞$
    Means " $x$ grows in the positive direction of ℝ".

  • $xâŸ¶ïżąâˆž$
    Means " $x$ grows in the negative direction of ℝ".

Example: Even Polynomials

  • When $x⟶∞$ then $𝐏_{2n}[x]⟶∞$
  • When $xâŸ¶ïżąâˆž$ then $𝐏_{2n}[x]⟶∞$

Example: Odd Polynomials

  • When $x⟶∞$ then $𝐏_{2n-1}[x]⟶∞$
  • When $xâŸ¶ïżąâˆž$ then $𝐏_{2n-1}[x]âŸ¶ïżąâˆž$

Then: The behavior of any polynomial $𝐏_{n}[x]$ corresponds to the highest non-zero term.