how to find horizontal asymptote of a function

To find the horizontal asymptote of a function, we look at the behavior of the function as the input \( x \) approaches infinity (\( x \to \infty \)) or negative infinity (\( x \to -\infty \)). Horizontal asymptotes represent the values that the function approaches as \( x \) grows larger in magnitude.

Here are the steps depending on the type of function:

### 1. **Rational Functions** (quotient of two polynomials):
For a rational function of the form:

\[
f(x) = \frac{a_n x^n + a_{n-1} x^{n-1} + \dots + a_0}{b_m x^m + b_{m-1} x^{m-1} + \dots + b_0}
\]

- Compare the **degree** of the numerator (\( n \)) and the degree of the denominator (\( m \)):

  - If \( n < m \): The horizontal asymptote is \( y = 0 \) (the x-axis).
  - If \( n = m \): The horizontal asymptote is \( y = \frac{a_n}{b_m} \), which is the ratio of the leading coefficients.
  - If \( n > m \): There is **no horizontal asymptote**, but there could be an **oblique (slant) asymptote**.

### 2. **Exponential Functions**:
For functions like \( f(x) = a e^{bx} + c \):

- If \( b > 0 \), the function grows without bound as \( x \to \infty \), but as \( x \to -\infty \), the function approaches the horizontal asymptote \( y = c \).
- If \( b < 0 \), the function decays as \( x \to \infty \), and the horizontal asymptote is \( y = c \).

### 3. **Logarithmic Functions**:
For logarithmic functions like \( f(x) = \log_b(x) \):

- Logarithmic functions do **not** have a horizontal asymptote since they grow slowly but without bound as \( x \to \infty \).

### 4. **Trigonometric Functions**:
For trigonometric functions like \( \sin(x) \) or \( \cos(x) \), these are periodic, so they do **not** have horizontal asymptotes.

### General Approach:
To find the horizontal asymptote for a given function, compute the limit of the function as \( x \) approaches infinity and negative infinity:

\[
\lim_{x \to \infty} f(x) \quad \text{and} \quad \lim_{x \to -\infty} f(x)
\]

If the limit exists and is a finite number, that value is the horizontal asymptote.

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### Example:
For \( f(x) = \frac{2x^2 + 3}{x^2 - 5} \):

- The degree of both the numerator and denominator is 2.
- So, the horizontal asymptote is \( y = \frac{2}{1} = 2 \).

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Do you want more details or have any questions?

Here are 5 related questions to deepen your understanding:
1. How do you find the horizontal asymptote for the function \( f(x) = \frac{3x^2 - 4x + 2}{x^3 + 5x^2 + 1} \)?
2. How do vertical asymptotes differ from horizontal asymptotes?
3. What is the horizontal asymptote of \( f(x) = 4e^{-x} + 5 \)?
4. How does one find an oblique (slant) asymptote?
5. Why don't trigonometric functions have horizontal asymptotes?

**Tip**: Always compare the degrees of the numerator and denominator when dealing with rational functions to quickly find the horizontal asymptote.

Learn how to find the horizontal asymptote of a function by analyzing limits at infinity. This guide covers rational functions, exponential functions, and other key concepts. Suitable for students in advanced high school or introductory calculus.

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