Understanding Horizontal Asymptotes in Mathematics | Exploring their Roles and Determination

Horizontal Asymptote

In mathematics, a horizontal asymptote is a horizontal line that a function approaches as its input (x) approaches positive or negative infinity

In mathematics, a horizontal asymptote is a horizontal line that a function approaches as its input (x) approaches positive or negative infinity. It represents the long-term behavior of the function as x becomes very large or very small.

To determine if a function has a horizontal asymptote, we often look at the limits of the function as x approaches infinity or negative infinity. If these limits exist and are equal to a constant value, then that constant value is the equation of the horizontal asymptote.

Let’s consider an example. Suppose we have the function f(x) = (2x + 1) / (x – 3). To determine if it has a horizontal asymptote, we calculate the limits as x approaches positive or negative infinity.

As x approaches infinity, the expression (2x + 1) / (x – 3) becomes dominated by the term with the highest power of x. In this case, it is the term 2x / x, which simplifies to just 2. Therefore, the limit as x approaches infinity is 2.

Similarly, as x approaches negative infinity, the expression (2x + 1) / (x – 3) also becomes dominated by the term 2x / x, which simplifies to 2. Thus, the limit as x approaches negative infinity is also 2.

Since the limits as x approaches infinity and negative infinity are equal to the constant value 2, the equation of the horizontal asymptote for the function f(x) = (2x + 1) / (x – 3) is y = 2.

This means that as x becomes very large or very small, the function f(x) gets closer and closer to the line y = 2 without actually reaching it. The role of the horizontal asymptote is to provide information about the behavior of the function at extreme x-values.

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