Problem 12 - 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.) $$ f(x)=\frac{1}{x+2} $$

Step-by-Step Solution

Answer

The horizontal and vertical asymptotes of the function \(f(x) = \frac{1}{x+2}\) are: - Vertical asymptote: \(x = -2\) - Horizontal asymptote: \(y = 0\)

Step 1: Find the vertical asymptote(s)

To find the vertical asymptotes, we need to determine where the function is undefined. This occurs when the denominator of the function is equal to zero. So, let's set the denominator equal to zero and solve for x: $$ x + 2 = 0 $$ From this equation, we can see that the function is undefined when \(x = -2\). Therefore, there is a vertical asymptote at \(x = -2\).

Step 2: Find the horizontal asymptote(s)

To find the horizontal asymptotes, we need to examine the limits of the function as x approaches positive or negative infinity. In this case, we will look at the following two limits: $$ \lim_{x \to \infty} \frac{1}{x + 2} $$ and $$ \lim_{x \to -\infty} \frac{1}{x + 2} $$ As x approaches infinity, the value of \(x + 2\) also approaches infinity, making the fraction approach zero: $$ \lim_{x \to \infty} \frac{1}{x + 2} = 0 $$ Similarly, as x approaches negative infinity, the value of \(x + 2\) also approaches negative infinity, but the fraction still approaches zero: $$ \lim_{x \to -\infty} \frac{1}{x + 2} = 0 $$ Since the limits in both cases approach zero, there is a horizontal asymptote at \(y = 0\). Now we have both the horizontal and vertical asymptotes of the given function \(f(x) = \frac{1}{x+2}\): - Vertical asymptote: \(x = -2\) - Horizontal asymptote: \(y = 0\)

Key Concepts

Horizontal Asymptotes

In analyzing functions, horizontal asymptotes represent the value a function approaches as the independent variable, often denoted by 'x', moves towards infinity or negative infinity. They are like the function's horizon, providing insight into the behavior of the graph far to the right (as x approaches positive infinity) or far to the left (as x approaches negative infinity).

To find these horizontal asymptotes, if they exist, we use the concept of limits. For the given function, \( f(x) = \frac{1}{x+2} \), we examine the limits \( \lim_{{x \to \infty}} \frac{1}{x + 2} \) and \( \lim_{{x \to -\infty}} \frac{1}{x + 2} \). In both cases, as 'x' becomes larger (positive or negative), the denominator grows without bounds, while the numerator remains constant at 1. This situation makes the overall value of the rational expression approach zero. Therefore, the graph of \()f\)able to easily predict the end behavior of a function's graph.

Vertical Asymptotes

Vertical asymptotes highlight the 'x' values where a function heads towards infinity or negative infinity; the function is essentially undefined at these points. These asymptotes are mostly associated with rational functions, of which our example \( f(x) = \frac{1}{x+2} \) is one.

To find a vertical asymptote, we zero in on the values that make the denominator of a rational function equal to zero, since division by zero is undefined. For the function we're discussing, setting the denominator \(x + 2 = 0\) reveals an asymptote at \(x = -2\). This means, as 'x' gets very close to -2, the function's value will grow very large in magnitude, shooting off towards infinity or negative infinity, thereby never touching the line x = -2 but coming arbitrarily close to it.

Limits of a Function

The concept of limits is fundamental in calculus and is used to describe the behavior of functions as they approach specific points or infinity. When dealing with a function, particularly a rational one, limits help us understand the value it reaches as the input gets arbitrarily close to a given number, or as it goes off towards infinity.

In simpler terms, limits answer the question: As we get closer and closer to a certain 'x' value, what value is the function trying to reach? For instance, in our example with \( f(x) = \frac{1}{x+2} \), we explored what happens as x goes to infinity (\( \lim_{x \to \infty} \frac{1}{x + 2} = 0 \) ) and negative infinity (\( \lim_{x \to -\infty} \frac{1}{x + 2} = 0 \)). By finding the limits, we were able to determine the horizontal asymptote at \(y = 0\).

Rational Functions

Rational functions are ratios of two polynomials where the denominator is not the zero polynomial. In the context of our example, \( f(x) = \frac{1}{x+2} \) is a simple rational function consisting of a polynomial of degree zero (the numerator) over a polynomial of degree one (the denominator).

Rational functions often exhibit interesting features such as asymptotes. A vertical asymptote occurs where the denominator equals zero (indicating a division by zero is happening), and horizontal asymptotes, if they exist, are discovered by evaluating the behavior of the function at the infinities. In understanding rational functions, students must get comfortable with both polynomials and the idea of division by expressions, to ensure they can tackle any asymptotic behavior and limits correctly.