How To Find Asymptotes Of Graphs - Singapore Maths Tuition

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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, $\displaystyle y=\frac{2}{x-3}$ . 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. $x-3=0$ , so $x=3$ .

graph1 — $\displaystyle y=\frac{2}{x-3}$

Another way vertical asymptotes can come about is via logarithmic graphs, e.g. $y=\ln (x+2)$ .

$\ln 0$ is undefined, so when $x+2=0$ or $x=-2$ , there will be a vertical asymptote at $x=-2$ .

Horizontal Asymptote

Horizontal asymptotes usually come about when one of the terms approaches zero as $x$ 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, $\displaystyle y=\frac{1}{x}+1$ . 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 $y=1$ .

graph3 — $\displaystyle y=\frac{1}{x}+1$

Another time where Horizontal Asymptotes appear is for Exponential Graphs. For instance, $y=e^{-x}+1$ . When x is very large, $e^{-x}$ will be very small, and hence $y$ approaches 1. This means that the Horizontal Asymptote will be $y=1$ .