Horizontal Asymptotes of Rational Function | GeoGebra

How to Find Horizontal Asymptotes of a Rational Function?

The rational function is defined as:

$f(x)={\displaystyle \frac{P(x)}{Q(x)}}$

and the rational function under considerations is:

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

The derivative of the given rational function is:

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

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

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

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:

$x=0,-0.946611\dots ,1.18754\dots$

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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