To find the horizontal asymptote of a function, you need to examine the behavior of the function as $x$ approaches infinity $(+\infty)$ or negative infinity $(-\infty)$. Here are the main cases to determine horizontal asymptotes based on the type of function:

1. Rational Functions (a function of the form $f(x) = \frac{P(x)}{Q(x)}$, where $P(x)$ and $Q(x)$ are polynomials):

• Case 1: Degree of the numerator = Degree of the denominator
  If the degree of the numerator and denominator are equal, the horizontal asymptote is found by dividing the leading coefficients of the highest degree terms in the numerator and denominator. $y = \frac{\text{leading coefficient of } P(x)}{\text{leading coefficient of } Q(x)}$

• Case 2: Degree of the numerator < Degree of the denominator
  If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is $y = 0$.

• Case 3: Degree of the numerator > Degree of the denominator
  If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote (instead, you may have an oblique asymptote or none).

2. Exponential Functions (functions like $f(x) = a \cdot b^x + c$):

• The horizontal asymptote is typically $y = c$, where $c$ is the value the function approaches as $x \to \infty$ or $x \to -\infty$.

3. Other Functions:

For trigonometric, logarithmic, and other types of functions, you may have to examine the specific form of the function and use limits as $x \to \infty$ or $x \to -\infty$ to determine the horizontal asymptote.

Example:

For the rational function $f(x) = \frac{2x^2 + 3x + 1}{x^2 - 4}$, the degrees of the numerator and denominator are both 2. The leading coefficients are 2 (from the numerator) and 1 (from the denominator), so the horizontal asymptote is: $y = \frac{2}{1} = 2$

Do you want further details or examples on how to find horizontal asymptotes?

Related Questions:

1. How do you find vertical asymptotes?
2. What are oblique asymptotes and when do they occur?
3. How do you use limits to find horizontal asymptotes?
4. Can logarithmic functions have horizontal asymptotes?
5. How does the horizontal asymptote differ in exponential decay versus exponential growth?

Tip:

For rational functions, always compare the highest powers of $x$ in both the numerator and denominator to quickly determine the horizontal asymptote.