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.