Determining Horizontal Asymptotes: A Guide to Understanding and Applying the Key Rules

Horizontal Asymptote Rules

Horizontal asymptotes are horizontal lines that a function approaches as the input values become large or small

Horizontal asymptotes are horizontal lines that a function approaches as the input values become large or small. They can provide valuable information about the behavior of a function when its input values are extremely large or extremely small.

There are three key rules to determine the horizontal asymptotes of a function:

1. Rule 1: Constant Term
If a function has a constant term (a number) in its expression, then the horizontal asymptote is the same as this constant term. In other words, if a function is of the form f(x) = c, where c is a constant, then the horizontal asymptote is y = c.

Example:
f(x) = 2
In this case, the horizontal asymptote is y = 2.

2. Rule 2: Degree of the Numerator and Denominator
Consider a rational function of the form f(x) = (p(x))/(q(x)), where both p(x) and q(x) are polynomials. If the degree of the denominator (q(x)) is greater than the degree of the numerator (p(x)), then the horizontal asymptote is y = 0.

Example:
f(x) = (5x^2 + 2)/(3x^4 – x^2)
Here, the degree of q(x) is 4 and the degree of p(x) is 2. Thus, the horizontal asymptote is y = 0.

3. Rule 3: Degree of the Numerator and Denominator (Continued)
If the degree of the numerator (p(x)) is equal to the degree of the denominator (q(x)), then the horizontal asymptote is given by the ratio of the leading coefficients of the numerator and denominator. The leading coefficient is the coefficient of the term with the highest degree in the polynomial.

Example:
f(x) = (4x^3 – 2)/(5x^3 + 3x^2 – 1)
Here, the leading coefficient of p(x) is 4 and the leading coefficient of q(x) is 5. Thus, the horizontal asymptote is y = 4/5.

It’s important to note that these rules only apply to rational functions (functions in the form of fractions). Other types of functions, such as exponential or trigonometric functions, may have different approaches to determine their horizontal asymptotes.

Remember that these rules provide a general guideline, but there may be cases where a function does not have a horizontal asymptote or approaches a different line. The behavior of a function near its horizontal asymptote can also be affected by other factors, such as vertical asymptotes or local extrema.

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