To find the horizontal asymptotes of a function, especially rational functions, follow these steps and rules:
General Method Using Limits
- Calculate the limit of the function as x→∞x\to \infty x→∞.
- Calculate the limit of the function as x→−∞x\to -\infty x→−∞.
- If either limit is a finite number kkk, then the horizontal asymptote is the line y=ky=ky=k
For Rational Functions f(x)=p(x)q(x)f(x)=\frac{p(x)}{q(x)}f(x)=q(x)p(x)
where ppp and qqq are polynomials:
- Let nnn be the degree of the numerator p(x)p(x)p(x).
- Let ddd be the degree of the denominator q(x)q(x)q(x).
Rules:
- If n<dn<dn<d, the horizontal asymptote is y=0y=0y=0.
- If n=dn=dn=d, the horizontal asymptote is y=leading coefficient of p(x)leading coefficient of q(x)y=\frac{\text{leading coefficient of }p(x)}{\text{leading coefficient of }q(x)}y=leading coefficient of q(x)leading coefficient of p(x).
- If n>dn>dn>d, there is no horizontal asymptote (there may be a slant asymptote instead)
Example:
For f(x)=2xx−3f(x)=\frac{2x}{x-3}f(x)=x−32x:
- Degree of numerator n=1n=1n=1
- Degree of denominator d=1d=1d=1
- Since n=dn=dn=d, horizontal asymptote is ratio of leading coefficients: y=21=2y=\frac{2}{1}=2y=12=2
For Exponential Functions:
- The horizontal asymptote is given by the vertical shift ccc in a function of the form f(x)=abkx+cf(x)=ab^{kx}+cf(x)=abkx+c.
- For example, f(x)=4x+2f(x)=4^x+2f(x)=4x+2 has horizontal asymptote y=2y=2y=2
Summary Table for Rational Functions
Condition on degrees| Horizontal Asymptote
---|---
n<dn<dn<d| y=0y=0y=0
n=dn=dn=d| y=leading coeff. numeratorleading coeff.
denominatory=\frac{\text{leading coeff. numerator}}{\text{leading coeff.
denominator}}y=leading coeff. denominatorleading coeff. numerator
n>dn>dn>d| No horizontal asymptote
This approach allows you to quickly identify horizontal asymptotes by comparing polynomial degrees or using limits for more general functions
