What is how to find the horizontal asymptote?

To find the horizontal asymptote of a function, you primarily need to analyze the function's behavior as x approaches positive or negative infinity. Here's a breakdown of the common scenarios:

1. Rational Functions (Polynomial/Polynomial): For a rational function, which is a function in the form of p(x)/q(x), where p(x) and q(x) are polynomials, compare the degrees of the numerator and the denominator.

   • Degree of numerator < Degree of denominator: The horizontal asymptote is y = 0 (the x-axis). See "[https://www.wikiwhat.page/kavramlar/degree%20of%20%a0polynomial](degree of polynomial)".
   • Degree of numerator = Degree of denominator: The horizontal asymptote is y = (leading coefficient of numerator) / (leading coefficient of denominator).
   • Degree of numerator > Degree of denominator: There is no horizontal asymptote. There may be a slant (oblique) asymptote, which you would find using polynomial long division.

2. Exponential Functions: Examine the behavior of the function as x approaches positive or negative infinity. Typically, exponential functions of the form f(x) = a^x (where a is a constant) will have a horizontal asymptote at y = 0 when x approaches negative infinity if a > 1, or as x approaches positive infinity if 0 < a < 1. Shifts and transformations can alter the location of the asymptote. See "[https://www.wikiwhat.page/kavramlar/exponential%20functions](exponential functions)".

3. Functions with Limits: The most general approach involves finding the limits:

   • lim_x→∞ f(x) = L (L is a constant). This indicates a horizontal asymptote at y = L.
   • lim_x→-∞ f(x) = M (M is a constant). This indicates a horizontal asymptote at y = M. Note that L and M can be different. If either limit approaches infinity, then there is no horizontal asymptote for that direction. See "https://www.wikiwhat.page/kavramlar/limits".

4. Logarithmic Functions: Logarithmic functions (like f(x) = log(x)) do not have horizontal asymptotes. Instead, they have a vertical asymptote. See "[https://www.wikiwhat.page/kavramlar/logarithmic%20functions](logarithmic functions)".

Important Considerations:

• A function can cross a horizontal asymptote. The horizontal asymptote describes the function's behavior at extreme values of x, not necessarily its behavior in the middle.
• Horizontal asymptotes describe end behavior.
• Use a graphing calculator or software to visualize the function and confirm your analytical results.