Question

How do you find the horizontal asymptote of a curve?

Answer 1

You will need to use the Limit Rules, the concepts of limits at infinity, and the following theorem to find the horizontal asymptote (generally of a rational function):
${\lim _{x \to \infty }}\dfrac{1}{{{x^r}}} = 0$
Where $r$ is rational and ${x^r}$ is defined.
And to apply this theorem, divide each term of the numerator and the denominator with the highest power term present at the denominator of the function.

Complete step by step solution:
In order to find the horizontal asymptote of a curve, we will use the method for using the limit theorem that is to divide each term from the denominator by the highest power term. This will leave us with a numerator polynomial or a constant. If we have a polynomial, no horizontal asymptote exists and if we have a constant, then our horizontal asymptote is $y = a,\;{\text{where}}\;a$ is the constant.
Let us take an example to understand this,
Find the horizontal asymptote of $f(x) = \dfrac{{2{x^3} - 3}}{{3{x^3} + 4x}}$
First of all, we will divide both numerator and denominator with ${x^3}$, we will get
$f(x) = \dfrac{{\dfrac{{2{x^3} - 3}}{{{x^3}}}}}{{\dfrac{{3{x^3} + 4x}}{{{x^3}}}}} = \dfrac{{2 - \dfrac{3}{{{x^3}}}}}{{3 + \dfrac{4}{{{x^2}}}}}$
Now, taking limits, we will get
${\lim _{x \to \infty }}f(x) = {\lim _{x \to \infty }}\dfrac{{\left( {2 - \dfrac{3}{{{x^3}}}} \right)}}{{\left( {3 + \dfrac{4}{{{x^2}}}} \right)}}$
Using properties of limit, we can further write it as
${\lim _{x \to \infty }}f(x) = \dfrac{{{{\lim }_{x \to \infty }}2 - {{\lim }_{x \to \infty }}\dfrac{3}{{{x^3}}}}}{{{{\lim }_{x \to \infty }}3 + {{\lim }_{x \to \infty }}\dfrac{4}{{{x^2}}}}} = \dfrac{{2 - 0}}{{3 + 0}} = \dfrac{2}{3}$
Therefore $y = \dfrac{2}{3}$ is the horizontal asymptote of the curve $f(x) = \dfrac{{2{x^3} - 3}}{{3{x^3} + 4x}}$

Note: If the highest degree of the denominator is greater than the highest degree of the numerator then $y = 0$ is the horizontal asymptote and when highest degree of numerator is greater than the highest degree of the denominator then there is no horizontal asymptote and when highest degrees of both are equal then asymptote is coefficient of highest degree term of numerator divided by coefficient of highest degree term of denominator.