How to Find Horizontal Asymptote
Calculating horizontal asymptotes is an important technique in analyzing the behavior of functions towards infinity or negative infinity. Here’s a step-by-step guide on how to find horizontal asymptotes:
Step 1: Analyzing the Function’s Degree
Start by determining the degrees of the numerator and denominator of the function. Let’s assume we have a rational function, where the numerator and denominator are both polynomials.
- If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote.
- If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is always at y = 0 (the x-axis).
- If the degrees of the numerator and denominator are equal, move to the next steps for further analysis.
Step 2: Dividing the Leading Terms
In this step, divide the leading term of the numerator by the leading term of the denominator. By doing this, you will obtain a horizontal asymptote.
- If the result of the division is a finite number, the horizontal asymptote is y = that number.
- If the result is infinity (∞) or negative infinity (-∞), there is no horizontal asymptote.
Step 3: Analyzing the Function’s Behavior
To further analyze the function’s behavior near the horizontal asymptote, compare the degrees of the highest power terms in the numerator and denominator. Analyze the following scenarios below:
- If the degree of the numerator is one more than the degree of the denominator, the function has a slant (oblique) asymptote. In this case, the horizontal asymptote found in step 2 is a never-touching boundary for the function.
- If the degree of the numerator is equal to the degree of the denominator, there may be a hole or a removable discontinuity at the value found in step 2. To determine if there is a hole, check for common factors between the numerator and denominator. If common factors exist, cancel them out and verify if a hole exists by substituting the found value into the simplified function.
By following these steps, you can easily find the horizontal asymptote(s) of a function. Remember, practice makes perfect, and verifying with graphs or further mathematical analysis is always recommended for a comprehensive understanding.