Question

True or False It is possible for a function to have more than one horizontal asymptote. Justify your answer.

Short answer

True. It is possible for a function to have more than one horizontal asymptote.

Step-by-step solution

Understanding the concept

To confirm or refute this statement, we need to understand what a horizontal asymptote is. A horizontal asymptote is a horizontal line of the form \(y = k\) where \(k\) is a real number, that the graph of a function gets ever closer to but never reaches, as the independent variable approaches positive or negative infinity.

Determining the possibility

Now, we determine if a function can have more than one horizontal asymptote. The answer is true; a function can have more than one horizontal asymptote. The reasoning behind this is because there can be one horizontal asymptote as x approaches positive infinity, and a different one as x approaches negative infinity.

Example

For instance, consider the function \(f(x) = (2x^2 + x - 1)/(x^2 +1)\). As \(x\) approaches negative infinity, the function behaves like \(f(x) = 2x^2/x^2 = 2\) and as \(x\) approaches positive infinity, the function behaves more like \(f(x) = 2x^2/x^2 = 2\). So, it has two horizontal asymptotes at \(y=2\).