Horizontal Asymptotes of Rational Function | GeoGebra

How to Find Horizontal Asymptotes of a Rational Function?

The rational function is defined as:

\(f(x)=\dfrac{P(x)}{Q(x)}\)

and the rational function under considerations is:

\(f(x)=\dfrac{(2x^4-x^3+7)}{(1+x^2)}\)

The derivative of the given rational function is:

\(\dfrac{d}{dx}f(x) = \dfrac{(4 x^5 - x^4 + 8 x^3 - 3 x^2 - 14x))}{(1 + x^2)^2}\)

Horizontal asymptotes are available to the given function whenever the slope i.e. derivative is zero, so by putting

\(\dfrac{(4 x^5 - x^4 + 8 x^3 - 3 x^2 - 14x))}{(1 + x^2)^2} = 0\)

We get, the roots of the polynomial available in the numerator. 3 roots are real in Nature while 2 are complex. The real roots are given as:

\(x = 0, -0.946611\ldots, 1.18754\ldots\)

Evaluating these roots in the given function \(f(x)\) and plot horizontal asymptotes of rational function in Geogebra

Complete Exercise in GeoGebra Worksheet

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