Question

In Exercises \(27-34,\) (a) find the vertical asymptotes of the graph of \(f(x) .(\) b) Describe the behavior of \(f(x)\) to the left and right of each vertical asymptote. $$f(x)=\frac{1}{x^{2}-4}$$

Short answer

The vertical asymptotes for the function \(f(x)=\frac{1}{x^{2}-4}\) are at \(x=2\) and \(x=-2\). The behaviour of \(f(x)\) is as follows: At \(x=2\), \(f(x)\) decreases to negative infinity from the left and increases to positive infinity from the right. At \(x=-2\), \(f(x)\) increases to positive infinity from the left and decreases to negative infinity from the right.

Step-by-step solution

Determination of Asymptotes

Vertical asymptotes occur where the denominator of a fraction is equal to zero (as division by zero is undefined). For this function, identify when \(x^{2}-4=0\) which will give the value of \(x\). Solving for \(x\), the equation becomes \(x^{2}=4\) and by taking the square root of both sides, the solutions obtained will be \(x=2\) and \(x=-2\) for the vertical asymptotes.

Describing the Left and Right Behavior around the Asymptotes

To describe the behavior of \(f(x)\) around each asymptote, calculate the limit of \(f(x)\) as \(x\) approaches the asymptote value from the left and the right. Note that when the denominator of the function is a quadratic or higher degree polynomial, the sign change rule of odd and even powers will apply. With the obtained asymptote values of \(x=2\) and \(x=-2\), the limits of \(f(x)\) will exhibit different behaviours when \(x\) approaches these values from the left and right respectively.

Calculating Limits and Finalizing Behaviours

To determine the behavior at \(x=2\), calculate the limit of \(f(x)\) as \(x\) approaches 2 from the left \(\lim_{x \to 2^-} f(x) = -\infty\) and from the right \(\lim_{x \to 2^+} f(x) = \infty\). Likewise, compute limit for \(x=-2\): be from the left \(\lim_{x \to -2^-} f(x) = \infty\) and from the right \(\lim_{x \to -2^+} f(x) = -\infty\). From these solutions, we can see that the function decreases towards \(x=2\) from the left and increases from the right, while at \(x=-2\), the function increases from the left and decreases from the right side.

Key concepts

Limits of Functions

Understanding the concept of limits is essential when studying calculus, particularly with functions that aren't well-behaved at every point. A limit describes the value that a function approaches as the input (or 'x' value) approaches some value. In the exercise provided, we examine the behavior of a function as it approaches specific points that are not part of the function's domain, namely the points where the function has vertical asymptotes.

To analyze limits effectively, students should be familiar with the following:

• The idea that a limit can approach a real number, infinity, or negative infinity.
• Notation such as \( \lim_{x \to a^+} f(x) \) and \( \lim_{x \to a^-} f(x) \) for describing limits from the right (a+) and from the left (a-), respectively.
• The understanding that a limit may not exist if the function approaches different values from the left and the right.

As demonstrated in the solution, calculating the limit involves investigating the behavior of the function around points where division by zero would occur. For the function \( f(x)=\frac{1}{x^{2}-4} \), the limit as \( x \) approaches 2 or -2 is essential in determining the function's behavior near these critical points.

Behavior of Rational Functions

Rational functions, like the one in our example \( f(x)=\frac{1}{x^{2}-4} \), can exhibit complex behaviors, especially near points where their denominators equate to zero. These functions are characterized by the presence of one polynomial divided by another polynomial. One aspect of their behavior is the presence of vertical asymptotes, which are lines that the function will approach but never touch.

Here are a few noteworthy points on rational functions:

• The denominator's roots are where we look for potential vertical asymptotes, as they signify points where the function is undefined.
• Between any two vertical asymptotes, the function will typically be continuous and follow a predictable pattern.
• As \( x \) approaches the location of a vertical asymptote, the function's value can shoot up towards infinity or drop towards negative infinity, depending largely on the input's direction and the function's structure.

The step-by-step solution illustrates the evaluation of limits on either side of the potential asymptotes to predict the function's behavior, revealing the directions in which the function tends to infinity or negative infinity.

Solving Quadratic Equations

To find vertical asymptotes in the function \( f(x)=\frac{1}{x^{2}-4} \) as addressed in the problem, we must solve the quadratic equation \( x^2 - 4 = 0 \). Quadratic equations are fundamental to algebra and have the general form \( ax^2 + bx + c = 0 \) where a, b, and c are constants, and \( a \) is not zero.

Students should keep in mind certain strategies for solving these equations:

• Factoring the quadratic if it is factorable.
• Applying the quadratic formula \( x=\frac{-b \pm \sqrt{b^2-4ac}}{2a} \) if factoring is not feasible or inconvenient.
• Completing the square as an alternative method, particularly when the quadratic is not easily factorable.
• Recognizing that the solutions (also called 'roots') to the quadratic equations are the points where the graph of the equation crosses the x-axis.

In this specific exercise, the quadratic \( x^2 - 4 \) is factorable as \( (x-2)(x+2) \) and yields the roots \( x=2 \) and \( x=-2 \) which correspond to the vertical asymptotes of the rational function.