Determining the Number and Behavior of Horizontal Asymptotes in Real-Valued Functions

How many horizontal asymptotes can a function have?

For a real-valued function, there can be up to two horizontal asymptotes

For a real-valued function, there can be up to two horizontal asymptotes.

To determine the number of horizontal asymptotes, we need to consider the behavior of the function as the input approaches positive or negative infinity.

1. No horizontal asymptotes: If the function grows or decreases without bound as the input approaches positive or negative infinity, then there are no horizontal asymptotes. For example, the function f(x) = x does not have any horizontal asymptotes because as x approaches positive or negative infinity, f(x) also grows without bound.

2. One horizontal asymptote: If the function approaches a fixed value (a horizontal line) as the input approaches positive or negative infinity, then there is one horizontal asymptote. For example, the function g(x) = 1/x has a horizontal asymptote at y = 0, as the function approaches 0 as x approaches positive or negative infinity.

3. Two horizontal asymptotes: If the function approaches different fixed values (two distinct horizontal lines) as the input approaches positive or negative infinity from different directions, then there are two horizontal asymptotes. For example, the function h(x) = (x^2 + 1)/(x^2 – 1) has two horizontal asymptotes: y = 1 and y = -1. As x approaches positive or negative infinity, the function approaches 1 or -1, respectively.

Note that in some cases, a function may not have any horizontal asymptotes, one horizontal asymptote, or two horizontal asymptotes. The exact number and values of the asymptotes depend on the behavior of the function as the input approaches infinity.

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