Question

Determine \(\lim _{x \rightarrow \infty} f(x)\) and \(\lim _{x \rightarrow-\infty} f(x)\) for the following rational functions. Then give the horizontal asymptote of \(f\) (if any). $$f(x)=\frac{4 x}{20 x+1}$$

Short answer

The limits as x approaches infinity and negative infinity for the given rational function are both \(\frac{1}{5}\), and the horizontal asymptote is at \(y=\frac{1}{5}\).

Step-by-step solution

Simplify the function

First, we can simplify the function by canceling out common factors. As 4 divides both the numerator and the denominator, we can simplify the function as follows: $$f(x)=\frac{4x}{4(5x+ \frac{1}{4})}$$ Cancel out the common factor of 4: $$f(x)=\frac{x}{5x+\frac{1}{4}}$$

Determine \(\lim _{x \rightarrow \infty} f(x)\)

As \(x \rightarrow \infty\), the dominating term in the denominator is 5x, whereas in the numerator, it is x. We can use this fact to find the limit as follows: $$\lim_{x \rightarrow \infty}\frac{x}{5x+\frac{1}{4}} \Rightarrow \lim_{x \rightarrow \infty}\frac{1}{5+\frac{1}{4x}}= \frac{1}{5+0}$$ Therefore, the limit as \(x\rightarrow \infty\) is: $$\lim _{x \rightarrow \infty} f(x)=\frac{1}{5}$$

Determine \(\lim _{x \rightarrow -\infty} f(x)\)

Similarly, as \(x \rightarrow -\infty\), the dominating term in the denominator is 5x, whereas in the numerator, it is x. We can use this fact to find the limit as follows: $$\lim_{x \rightarrow -\infty}\frac{x}{5x+\frac{1}{4}} \Rightarrow \lim_{x \rightarrow -\infty}\frac{1}{5+\frac{1}{4x}}= \frac{1}{5-0}$$ Therefore, the limit as \(x\rightarrow -\infty\) is: $$\lim _{x \rightarrow -\infty} f(x)=\frac{1}{5}$$

Determine the horizontal asymptote

Since both the limits at infinity and negative infinity for the function are equal, a horizontal asymptote exists for this function. The horizontal asymptote is a horizontal line with the equation: $$y=\frac{1}{5}$$ In conclusion, the limits as \(x \rightarrow \infty\) and \(x \rightarrow -\infty\) for the given rational function are both \(\frac{1}{5}\), and the horizontal asymptote is at \(y=\frac{1}{5}\).

Key concepts

Horizontal Asymptote

When studying the behavior of rational functions as the variable approaches infinity, we often encounter the term horizontal asymptote. This term refers to a horizontal line that the graph of the function approaches but never actually reaches as the variable gets arbitrarily large in either the positive or negative direction.

In our given function, after simplifying and analyzing the limits as x approaches positive and negative infinity, we determined that the horizontal asymptote is represented by the equation y = 1/5. An easy way to find the horizontal asymptote for a given rational function is to compare the degrees of the numerator and the denominator. If the degrees are the same, the horizontal asymptote is the ratio of the coefficients of the highest degree terms.

Limit at Infinity

Understanding the limit at infinity is crucial when you want to know the end behavior of a function. In essence, this limit tells you what value the function is approaching as the input grows larger and larger without bounds.

The function f(x) = 4x / (20x + 1) gives us a clear illustration of this concept. We saw that as x tends to positive or negative infinity, the terms containing x dominate the behavior of the function. Consequently, the function approaches a fixed value, 1/5, which is also the horizontal asymptote. For rational functions, the limit at infinity is often determined by the dominating terms of the numerator and the denominator.

Simplifying Rational Expressions

Before solving for limits or asymptotes, it's beneficial to simplify rational expressions. This involves reducing common factors and rewriting the function in a simpler form, making it easier to identify key characteristics like horizontal asymptotes or limits at infinity.

In the provided exercise, the original function, f(x) = 4x / (20x + 1), was simplified by dividing both the numerator and the denominator by 4, removing a common factor. Simplification may reveal hidden patterns and aid in the visualization of the function's behavior, laying the groundwork for further analysis such as determining horizontal asymptotes.

Dominating Term Analysis

The concept of dominating term analysis is a method used to find limits at infinity for rational functions. It involves identifying the term in both the numerator and the denominator that grows at the fastest rate as the variable becomes very large or very small.

In the given function, the term x in the numerator and 5x in the denominator are the dominating terms. When x goes to infinity or negative infinity, the other terms in the function become negligible compared to these dominating terms. Thus, analyzing the behavior of these terms alone suffices to establish the limits and horizontal asymptote of the function, as evidenced by the steps taken in solving the original exercise.