Determining Horizontal and Vertical Asymptotes | Steps and Methods for Rational Functions

How do I find a horizontal asymptote? How do I find a vertical asymptote?

To find a horizontal asymptote, you need to determine the behavior of a function as it approaches positive or negative infinity

To find a horizontal asymptote, you need to determine the behavior of a function as it approaches positive or negative infinity. Horizontal asymptotes can be determined by following these steps:

1. Simplify the function as much as possible.
2. Identify the highest power term in the numerator and the highest power term in the denominator.
3. If the highest power term in the numerator has a greater degree than the highest power term in the denominator, there is no horizontal asymptote. The function will have an oblique asymptote (slanted line).
4. If the highest power term in the numerator has the same degree as the highest power term in the denominator, the horizontal asymptote is a horizontal line with the equation y = (the coefficient of the highest power term in the numerator) / (the coefficient of the highest power term in the denominator).
5. If the highest power term in the denominator has a greater degree than the highest power term in the numerator, the horizontal asymptote is the x-axis (y = 0).

To find a vertical asymptote, you need to determine the behavior of the function as it approaches a specific value of x. Vertical asymptotes can be found by following these steps:

1. Simplify the function as much as possible.
2. Look for any values of x that make the denominator equal to zero.
3. Set the denominator equal to zero and solve for x. The values of x that satisfy this equation are the potential vertical asymptotes.
4. Check if the function has any removable discontinuities by simplifying the function further or canceling common factors in the numerator and denominator.
5. If there are no removable discontinuities, any vertical asymptotes occur at the values of x found in step 3.

It’s important to note that these methods apply to rational functions, which are functions where the numerator and denominator are both polynomials. Other types of functions may require different methods to find asymptotes.

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