Problem 37 - Essential Calculus

Question

In Exercises \(27-38,\) (a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hôpital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). \(\lim _{x \rightarrow 2^{+}}\left(\frac{8}{x^{2}-4}-\frac{x}{x-2}\right)\)

Step-by-Step Solution

Answer

The Type of indeterminate form is '0/0' and the value of the limit as \(x\) approaches 2 from the positive side is 0.

Step 1: Identify the indeterminate form

By direct substitution, replacing \(x\) with 2, the equation becomes \(\frac{8}{(2)^{2}-4} - \frac{2}{2-2}\), which equals \(\frac{8}{4-4} - \frac{2}{2-2}\). Both fractions give the forms of ‘0/0’ and are known as indeterminate forms.

Step 2: Apply L'Hôpital's Rule

To solve this problem, differentiate the numerator and denominator separately for each fraction. After differentiation, the equation becomes \(\lim _{x \rightarrow 2^{+}}\left(\frac{0}{2x} - \frac{(x^{2} -2x)}{(x-2)^2}\right)\). Again replace \(x\) with 2 in the equation, which equals \(\frac{0}{4} - \frac{(2^{2}- 2*2)}{(2-2)^2}\), simplifying this gives us 0.

Step 3: Verify the result graphically

Graph the functions \(y=\frac{8}{x^{2}-4}\) and \(y=\frac{x}{x-2}\) separately. Then subtract the second function from the first to get a plot of the given limit equation. By observing the graph as \(x\) approaches 2 from the positive side, you can confirm that it also converges towards 0.

Key Concepts

Understanding Indeterminate Forms

Indeterminate forms occur when you attempt to evaluate a limit and the result isn't immediately clear, like getting a \(\frac{0}{0}\) form. In mathematical terms, indeterminate means that the limit could potentially be any value. This form suggests that further analysis is necessary to determine the actual value of the limit. These situations are common in calculus when dealing with limits, especially when direct substitution leads to a nonsensical outcome.

• Indeterminate forms can appear in several types, such as \(\frac{0}{0}\), \(\frac{\infty}{\infty}\), \(0 \, \times \, \infty\), and more.
• Each type of indeterminate form has specific techniques for resolving them, like L'Hôpital's Rule, algebraic manipulation, or factoring.

In the given exercise, substituting \(x\) by 2 results in the \(\frac{0}{0}\) form, indicating the need for further work to evaluate this limit.

The Role of Limits in Calculus

A limit helps us understand how a function behaves as it approaches a certain point, either from the right, the left, or both directions. It's a fundamental concept in calculus that facilitates the study of continuity, derivatives, and integrals. Calculating limits is all about finding the value that a function approaches as its input gets closer to a particular point.

• Limits can be finite or infinite, depending on how the function behaves near the target value.
• If directly substituting the target value into the function results in a valid number, then that number is the limit.
• If not, techniques like factoring, rationalizing, or applying L'Hôpital's Rule might be useful.

In the problem given, after identifying the indeterminate form, L'Hôpital's Rule is applied, helping us efficiently evaluate the limit, which here simplifies to 0 when substituting \(x = 2\) again after differentiating.

Graphing Utilities for Visual Verification

Graphing utilities are tools that help visualize complex mathematical functions and aid in verifying calculations, like the limit evaluated through algebraic methods. They provide a graphical representation of the function's behavior as it approaches a certain point, offering an intuitive confirmation of calculated limits.

• These tools can graph the function, and their features often allow you to zoom in to closely inspect behavior near specific points.
• Using graphing utilities can reveal the nature of any discontinuities or asymptotic behaviors.

For the exercise, after evaluating the limit algebraically, these utilities show that as \(x\) approaches 2 from the positive side, the subtraction of the two functions \( \frac{8}{x^{2}-4} \) and \( \frac{x}{x-2} \) flattens out, confirming a limit value of 0. In this manner, graphing utilities enhance our understanding by providing a visual context to abstract mathematical results.