Option 10: Horizontal Asymptotes (Limits whose x-value is heading towards infinity)

Definition: Horizontal Asymptotes

The horizontal asymptote of an equation is the long-term behavior of an equation. So, for very, very, very large values of x (i.e. $\mathit{\pm }\mathit{\infty }$ ) , what $y–\mathit{value}$ does it look like your equation would head towards.

You find horizontal asymptotes by taking the limit of an equation as x approaches $\mathbf{+}\mathbf{\infty }$, and as x approaches $\mathit{–}\mathit{\infty }$.

$\underset{x\to \mathrm{\infty }}{\lim }f\left(x\right)=\mathit{Horizontal\; Asymptote}$ and $\mathrm{ }\underset{x\to –\mathrm{\infty }}{\lim }f\left(x\right)=\mathit{Horizontal\; Asymptote}$

Note:

–           The answer to a horizontal asymptote question is always the $y–\mathit{value}$ that the equation is heading towards.

–           You can have at most 2 horizontal asymptotes. One heading towards $\mathbf{+}\mathbf{\infty }$ and one heading towards $\mathit{–}\mathit{\infty }$, and they are not always the same value.

–           Your equation is allowed to cross the horizontal asymptote.