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How to Find a Horizontal Asymptote

What is a horizontal asymptote? What makes a graph have one, and how do you find one? These are some questions I got recently.

In this case, the end behavior is [latex]f\left(x\right)\approx \frac{4x}{{x}^{2}}=\frac{4}{x}[/latex]. This tells us that, as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\left(x\right)=\frac{4}{x}[/latex], and the outputs will approach zero, resulting in a horizontal asymptote at y = 0. Note that this graph crosses the horizontal asymptote.

In this case, the end behavior is [latex]f\left(x\right)\approx \frac{3{x}^{2}}{x}=3x[/latex]. This tells us that as the inputs increase or decrease without bound, this function will behave similarly to the function [latex]g\left(x\right)=3x[/latex]. As the inputs grow large, the outputs will grow and not level off, so this graph has no horizontal asymptote. However, the graph of [latex]g\left(x\right)=3x[/latex] looks like a diagonal line, and since f will behave similarly to g, it will approach a line close to [latex]y=3x[/latex]. This line is a slant asymptote. (Source: courses.lumenlearning.com)

A removable discontinuity occurs in the graph of a rational function at [latex]x=a[/latex] if a is a zero for a factor in the denominator that is common with a factor in the numerator. We factor the numerator and denominator and check for common factors. If we find any, we set the common factor equal to 0 and solve. This is the location of the removable discontinuity. This is true if the multiplicity of this factor is greater than or equal to that in the denominator. If the multiplicity of this factor is greater in the denominator, then there is still an asymptote at that value.

While vertical asymptotes describe the behavior of a graph as the output gets very large or very small, horizontal asymptotes help describe the behavior of a graph as the input gets very large or very small. Recall that a polynomial’s end behavior will mirror that of the leading term. Likewise, a rational function’s end behavior will mirror that of the ratio of the leading terms of the numerator and denominator functions. (Source: courses.lumenlearning.com)