2.2 - Limits Involving Infinity

• All Properties of Limits apply to limits approaching $\pm \mathrm{\infty }$ as does The Sandwich Theorem.

Asymptotes

In the graph of the function $y=f(x)$:

• $y=b$ is a horizontal asymptote if either:
  • ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}f(x)=b}$
  • ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}f(x)=b}$
• $x=a$ is a vertical asymptote if either:
  • ${\displaystyle \underset{x\to a^{+}}{lim}f(x)=\pm \mathrm{\infty }}$
  • ${\displaystyle \underset{x\to a^{-}}{lim}f(x)=\pm \mathrm{\infty }}$

End behavior models

• Small parts of the function become insignificant as inputs to it increase in size.
• So end behavior models (EBMs) are calculated using the leading terms of the numeration and denominator
  • example: EBM of $\frac{2x^5+x^4-x^2+1}{3x^2-5x+7}$ is simply $\frac{2x^5}{3x^2}$ or $\frac{2x^3}{3}$.

Formal definition of an end behavior model

The function $g$ is a…

• right-hand EBM for $f$ if and only if ${\displaystyle \underset{x\to \mathrm{\infty }}{lim}\frac{f(x)}{g(x)}=1}$
• left-hand EBM for $f$ if and only if ${\displaystyle \underset{x\to -\mathrm{\infty }}{lim}\frac{f(x)}{g(x)}=1}$
  Intuitively, this makes sense, because what this is really saying is that $f(x)$ and $g(x)$ need to be the same as you keep “zooming out” the graph.