Problem 15 - 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.) $$ f(x)=\frac{x-1}{x+1} $$

Step-by-Step Solution

Answer

The horizontal asymptote of the function \(f(x) = \frac{x-1}{x+1}\) is the line \(y=1\), and the vertical asymptote is the line \(x=-1\).

Step 1: Find the horizontal asymptote

To find the horizontal asymptote, we will need to examine the limit of the function as \(x\) approaches infinity. To do this, we will first rationalize the function by dividing both the numerator and the denominator by x. $$ f(x)=\frac{x-1}{x+1} = \frac{(x-1)/x}{(x+1)/x} $$ Now, we will simplify the function. $$ f(x)=\frac{1-\frac{1}{x}}{1+\frac{1}{x}} $$ Next, we will find the limit of the function as \(x\) approaches infinity. $$ \lim_{x\to\infty} f(x) =\frac{1-\frac{1}{x}}{1+\frac{1}{x}} $$ As \(x\) approaches infinity, the terms \(\frac{1}{x}\) in both the numerator and the denominator will approach 0. $$ \lim_{x\to\infty} \frac{1-\frac{1}{x}}{1+\frac{1}{x}} = \frac{1-0}{1+0} = 1 $$ Since the limit is equal to 1, the horizontal asymptote is the line \(y=1\).

Step 2: Find the vertical asymptote (if any)

To find a vertical asymptote, we need to look for any value of \(x\) that would cause the denominator of the function to be equal to zero. If there is such a value, the function will be undefined at this point, and this will result in a vertical asymptote. Let's solve the denominator equation \(x+1=0\) for x: $$ x+1 = 0\\ x = -1 $$ Since the denominator of the function is equal to zero when \(x=-1\), there will be a vertical asymptote at the line \(x=-1\). In summary: The horizontal asymptote of the function \(f(x)=\frac{x-1}{x+1}\) is the line \(y=1\), and the vertical asymptote is the line \(x=-1\).

Key Concepts

Horizontal Asymptote

Understanding horizontal asymptotes involves examining the behavior of a function as the input, \(x\), moves towards positive or negative infinity. The idea is to figure out what value \(f(x)\) is approaching or "flattening out to" as \(x\) becomes very large or very small. For a function like \(f(x) = \frac{x-1}{x+1}\), you consider the highest degree terms of the numerator and the denominator. Here, both have degree 1 (the terms are both \(x\)). Keep in mind:

• If the degrees of the numerator and denominator are equal, the horizontal asymptote is at \(y = \frac{a}{b}\), where \(a\) and \(b\) are the coefficients of the highest degree terms.
• If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at \(y = 0\).
• If the degree of the numerator is greater, there is no horizontal asymptote, but rather an oblique asymptote.

For our function, \(a\) and \(b\) are both 1, so the horizontal asymptote is \(y = 1\). This means as \(x\) approaches infinity, \(f(x)\) gets closer and closer to 1, never quite touching it, but hovering endlessly close.

Vertical Asymptote

A vertical asymptote occurs when \(f(x)\) tends to infinity as \(x\) approaches a certain value within its domain, typically where the denominator of a fraction equals zero. This results in the function being undefined at that point. To find vertical asymptotes of the function \(f(x) = \frac{x-1}{x+1}\), look directly at the equation of the denominator. Here, solve \(x+1=0\) which gives \(x = -1\).

• The presence of a vertical asymptote is confirmed at \(x = -1\), because it causes the denominator to become zero (making the function undefined).
• Vertical asymptotes commonly indicate a point of discontinuity in the graph where a function shoots up to infinity or drops to negative infinity.

A handy tip: always check the domain for any restrictions that might create these vertical lines and be cautious about points where the function is not defined.

Limits in Calculus

Limits are fundamentally about understanding the behavior of functions as they approach certain points or stretch out towards infinity. They're the backbone of both asymptote concepts discussed in this exercise. The limit analyses you performed determine where the graph of \(f(x)\) stays close to certain values without actually reaching them, leading to asymptotes.

• The limit \(\lim_{x \to \infty} f(x)\) helps identify the horizontal asymptote by showing what value \(f(x)\) trends towards as \(x\) becomes very large or very negative.
• Vertical asymptotes utilize limits too, often evaluated by considering \(\lim_{x \to a^\pm} f(x)\), where "\(a\)" is the potential asymptote and "\(\pm\)" indicates approaching from either side.

Limits are frequently tested around points where an asymptote might exist. Understanding how to effectively compute limits empowers you to identify places where a function behaves unusually, such as at infinity or undefined points, giving insight into the function's asymptotic nature.