Problem 11 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Find the horizontal and vertical asymptotes of the graph of the function. (You need not sketch the graph.) $$ f(x)=\frac{1}{x} $$

Step-by-Step Solution

Answer

The horizontal asymptote is \(y = 0\), and the vertical asymptote is \(x = 0\).

Step 1: Find Horizontal Asymptotes

We look at the behavior of the function as \(x\) approaches positive or negative infinity: 1. As \(x\) approaches positive infinity (\(x \to \infty\)), we can see that \(f(x) = \frac{1}{x}\) tends to zero (the fraction becomes smaller and smaller). Thus, the graph approaches the line \(y = 0\) as \(x \to \infty\). 2. Similarly, as \(x\) approaches negative infinity (\(x \to -\infty\)), we can see that \(f(x) = \frac{1}{x}\) also tends to zero (the fraction becomes smaller and smaller). Thus, the graph approaches the line \(y = 0\) as \(x \to -\infty\). Since the graph approaches the line \(y = 0\) as \(x\) goes to positive or negative infinity, we can conclude that there is a horizontal asymptote at \(y = 0\).

Step 2: Find Vertical Asymptotes

To find the vertical asymptote, we look for values of \(x\) that make the denominator, \(x\), of the function equal to zero. For the given function, \(f(x) = \frac{1}{x}\), the denominator is zero when \(x = 0\). So, we have a vertical asymptote at \(x = 0\). In conclusion, we found the horizontal asymptote at \(y = 0\) and the vertical asymptote at \(x = 0\).

Key Concepts

Horizontal Asymptote

Understanding horizontal asymptotes is crucial for visualizing how a function behaves as the input values become very large or very small. A horizontal asymptote of a graph is a horizontal line that the graph approaches but never actually reaches, even as we extend the graph towards infinity or negative infinity.

For the function given in the exercise, \( f(x) = \frac{1}{x} \), as the value of \( x \) increases or decreases without bound, the value of the function approaches zero. This means that the line \( y = 0 \) serves as a horizontal asymptote. Our analysis for horizontal limits shows that the graph will get closer and closer to this line, but won't intersect it, which characterizes the end behavior of the function.

Vertical Asymptote

A vertical asymptote represents a value of \( x \) where the function tends towards positive or negative infinity. Specifically, it's when the function cannot produce a value for a particular \( x \), indicating a kind of 'boundary' that the graph doesn't cross. In terms of the given function \( f(x) = \frac{1}{x} \), when \( x = 0 \), the function is undefined because division by zero is not allowed in mathematics.

The graph of \( f(x) \) will shoot upwards or downwards as it approaches \( x = 0 \), but will never cross this line. This is why we have a vertical asymptote at this point. So whenever you're looking for vertical asymptotes, remember, you're seeking the points where the denominator equals zero and the function is undefined.

Limit of a Function

The concept of a limit is at the heart of calculus and helps to describe the behavior of functions as they approach a certain point or infinity. It answers the question, 'What value does the function get close to as the inputs get close to a specified value?'

For \( f(x) = \frac{1}{x} \), we evaluate the limit as \( x \) approaches infinity and negative infinity to identify the horizontal asymptote. The limits, in this case, are both zero, simply meaning that as \( x \) gets larger in magnitude, the value of \( f(x) \) gets closer and closer to zero. These limit concepts reinforce what we have identified in our horizontal asymptote analysis.

Behavior of Rational Functions

Rational functions are quotients of polynomial functions. Their behavior is fascinating because they can exhibit both vertical and horizontal asymptotes. The function \( f(x) = \frac{1}{x} \) is a simple example of a rational function. Its behavior changes distinctly around the vertical asymptote and flattens out towards the horizontal asymptote.

As \( x \) approaches zero from either side, the values of the function become very large in magnitude (they either go towards positive or negative infinity, depending on the direction from which zero is approached). Yet, as \( x \) moves away from zero towards either positive or negative infinity, the value of the function gets smaller and closer to zero. This dual behavior around different asymptotes is a hallmark of rational functions and important to understand when analyzing their graphs.