This post is all about finding **Vertical and ****Horizontal **asymptotes of graphs.

## Vertical Asymptotes

Usually, vertical asymptotes come about when there is a rational function with a numerator and a denominator, for instance, . When the denominator is 0, the function is undefined, and hence there is a vertical asymptote there.

Hence, to find the asymptote, let the denominator be 0. E.g. , so .

Another way vertical asymptotes can come about is via logarithmic graphs, e.g. .

is undefined, so when or , there will be a vertical asymptote at .

## Horizontal Asymptote

Horizontal asymptotes usually come about when one of the terms approaches zero as approaches infinity.

To find the Horizontal Asymptote, find the value of y when x approaches infinity (i.e. when x becomes a very big number).

For example, . When x is a very big number, say x=10000, y will be close to 1 since 1/10000 is almost zero. Hence, the horizontal asymptote is .

Another time where Horizontal Asymptotes appear is for Exponential Graphs. For instance, . When x is very large, will be very small, and hence approaches 1. This means that the Horizontal Asymptote will be .

Note: The graphs above were drawn using the software Geogebra. 🙂

Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra