Question

Numerical and Graphical Analysis In Exercises 3-6, determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches -3 from the left and from the right by completing the table. Use a graphing utility to graph the function to confirm your answer. $$ f(x)=\frac{x^{2}}{x^{2}-9} $$

Short answer

As \(x\) approaches -3 from the right, \(f(x)\) approaches \(-\infty\). On the other hand, as \(x\) approaches -3 from the left, \(f(x)\) approaches \(\infty\).

Step-by-step solution

Understanding the Given Function

Given function is a rational function. When the denominator of a function approaches 0, the value of the function tends to infinity or negative infinity. He we observe that when \(x\) approaches to \(-3\) from either side in \(f(x)=\frac{x^{2}}{x^{2}-9}\), denominator will approach towards 0 (since \(-3^{2}-9 = 0\)). Now the function will tend to \(\infty\) or \(-\infty\) depending on the sign of denominator. Let's investigate this further.

Analyze Function when \(x\) approaches -3 from the right

To solve this, evaluate the function when \(x\) slightly greater than -3 (such as -2.999). Substitute the value in the function; \(f(x) = \frac{(-2.999)^{2}}{(-2.999)^{2} - 9}\). This will yield a negative value because the numerator is positive (as square of any real number is positive) and denominator is negative (as it is slightly smaller than 9). Thus, as \(x\) approaches -3 from the right, \(f(x)\) approaches \(-\infty\).

Analyze Function when \(x\) approaches -3 from the left

Now evaluate the function when \(x\) is slightly less than -3 (such as -3.001). Substitute this value in the function; \(f(x) = \frac{(-3.001)^{2}}{(-3.001)^{2} - 9}\). The result will be positive because both the numerator and denominator are positive. Thus, as \(x\) approaches -3 from the left, \(f(x)\) approaches \(\infty\).

Conclusion and Graphical Analysis

As \(x\) approaches -3 from the left and from the right, \(f(x)\) approaches \(\infty\) and \(-\infty\) respectively. A graph can confirm these findings. The graph would show the function approaching \(\infty\) as \(x\) approaches -3 from the left and the function approaching \(-\infty\) as \(x\) approaches -3 from the right.

Key concepts

Analyzing Limits Graphically

Visualizing the behavior of functions as they approach certain points can greatly enhance our understanding of their limits. For the function \( f(x) = \frac{x^{2}}{x^{2} - 9} \), graphing it helps us see what happens as we approach \( x = -3 \). A graphing utility can plot the function and clearly show that the function does not converge to a single value as \( x \) gets infinitely close to -3. Instead, the graph would depict two different behaviors on each side of the vertical line \( x = -3 \), which indicates the presence of a vertical asymptote.

The graphical representation would show that as \( x \) approaches -3 from the left, the function's output grows without bound, visualized by the graph extending upwards towards infinity. Conversely, from the right side, the output decreases without limit, depicted by the graph plunging downwards. Through this visualization, the concept of limits can be more intuitive, allowing students to predict the behavior of functions near critical points without solely relying on algebraic manipulation.

Asymptotic Behavior

When talking about the limit of a rational function, we often encounter the term 'asymptotic behavior.' This term describes the behavior of a function as it gets close to a certain line without ever touching it. In our specific example with the function \( f(x) = \frac{x^{2}}{x^{2} - 9} \), the line we are referring to is \( x = -3 \). As \( x \) approaches this value, the denominator of the function approaches zero, which in turn causes the value of the function to become unbounded—meaning it tends towards infinity or negative infinity. This is characteristic of a vertical asymptote.

The function exhibits different asymptotic behaviors on each side of \( x = -3 \). If we were to plot this on a graph, we would observe the function stretching infinitely upwards to the left of the vertical asymptote and downwards to the right. Understanding asymptotic behaviors is crucial for analyzing the end behavior of functions and determining their limits at points of discontinuity.

Rational Function Behavior

Rational functions display a variety of behaviors depending on the relation between the numerator and the denominator. The function \( f(x) = \frac{x^{2}}{x^{2} - 9} \) showcases a behavior that reflects the concept of limits in rational functions. Specifically, as \( x \) approaches values that make the denominator zero, we expect the function to demonstrate extreme behavior—either shooting up to infinity or plunging down to negative infinity.

Our earlier analysis displayed two distinct responses when \( x \) approaches -3. When approaching from the left, the function approaches infinity because the numerator and denominator are both positive, making the overall value of the function positive and large. On the other hand, approaching from the right yields a negative result since the square of a number just less than -3 will still be positive, but subtracting 9 would make the denominator negative. Thus, we get a negative result that gets more negative as we get closer to -3, leading to negative infinity. By examining these behaviors, students can get a better grasp on the nature of rational functions near critical points and how to calculate their limits.