What are good examples of functions with horizontal asymptotes of 5?
To find examples of functions with a horizontal asymptote of 5, we need to look for functions that approach 5 as the input (x) goes to positive or negative infinity. Here are a few common examples:
1. Constant Function: The function f(x) = 5 is a simple and basic example. It is a horizontal line that is always at a height of 5, and it approaches 5 as x goes to positive or negative infinity.
2. Rational Functions: Rational functions are ratios of polynomials and can have horizontal asymptotes. For example, consider the function f(x) = 5x / (x + 1). As x approaches infinity, the x term becomes dominant, and the function approaches 5. Similarly, as x approaches negative infinity, the x term is also dominant, and the function approaches 5.
3. Exponential Functions: Exponential functions in the form of f(x) = a^x, where a is a positive constant greater than 1, can have horizontal asymptotes. For instance, if we choose a = e (approximately 2.71828, Euler's number), then the function f(x) = e^x approaches 5 as x goes to negative infinity. This is because the exponential growth becomes negligible compared to the constant value 5.
These are just a few examples, but there are many other types of functions that can have a horizontal asymptote of 5. It's important to note that the behavior of a function as x approaches infinity may differ from its behavior as x approaches negative infinity, so it's worth exploring different types of functions to understand their asymptotic behavior.