Problem 272 - Calculus Volume 1

Question

Find the horizontal and vertical asymptotes. \(f(x)=\frac{1}{1-x^{2}}\)

Step-by-Step Solution

Answer

Vertical asymptotes at \(x = 1\) and \(x = -1\); horizontal asymptote at \(y = 0\).

Step 1: Identify Vertical Asymptotes

Vertical asymptotes occur at values of \(x\) where the denominator of a rational function is zero and the numerator is not zero. Set the denominator equal to zero: \[1 - x^2 = 0\] This simplifies to \[x^2 = 1\] Solving for \(x\), we get \[x = \pm 1\]. Thus, \(x = 1\) and \(x = -1\) are the vertical asymptotes.

Step 2: Identify Horizontal Asymptotes

To find horizontal asymptotes, compare the degrees of the numerator and the denominator of the function. The function \(f(x) = \frac{1}{1-x^2}\) has a numerator degree of 0 and a denominator degree of 2.If the degree of the denominator is greater than the degree of the numerator, the horizontal asymptote is \(y = 0\). Therefore, \(y = 0\) is the horizontal asymptote for this function.

Key Concepts

Vertical Asymptotes

In rational functions like \( f(x) = \frac{1}{1-x^2} \), vertical asymptotes are points where the function approaches infinity or negative infinity. They occur when the denominator equals zero, making the function undefined, while the numerator remains nonzero.
To find vertical asymptotes, set the denominator equal to zero. Here, solving \( 1 - x^2 = 0 \) gives us \( x^2 = 1 \). The solutions \( x = \pm 1 \) tell us that vertical asymptotes occur at \( x = 1 \) and \( x = -1 \).
What's happening graphically is that the function values increase or decrease without bound as \( x \) approaches these points. Understanding vertical asymptotes helps in graph sketching and understanding behavior near undefined points.

Horizontal Asymptotes

Horizontal asymptotes describe how a function behaves as \( x \) moves towards positive or negative infinity. For the rational function \( f(x) = \frac{1}{1-x^2} \), horizontal asymptotes depend on the degrees of the numerator and the denominator.
If the degree of the denominator (2 in this case) is greater than the numerator's degree (which is 0), the horizontal asymptote is \( y = 0 \). This means, as \( x \) stretches towards infinity in either direction, the function value will get closer and closer to 0.
Horizontal asymptotes don't indicate an unreachable line but rather suggest the end behavior trend of the graph. They help predict behavior at the extremes of the function's domain.

Rational Functions

Rational functions are expressions formed by dividing one polynomial by another. They are represented in the form \( f(x) = \frac{p(x)}{q(x)} \). An example is \( f(x) = \frac{1}{1-x^2} \). Here, \( p(x) = 1 \) and \( q(x) = 1-x^2 \).
Key characteristics include:

• Vertical asymptotes where the denominator is zero and the numerator isn't.
• Horizontal asymptotes determined by comparing the highest degrees of numerator and denominator polynomials.

Understanding these features is important as they determine the overall shape and behavior of the graph of the rational function. Recognizing and predicting asymptotic behavior is crucial for graphing and interpreting how these functions behave.