Problem 63 - Thomas Calculus

Question

In Exercises \(61-66,\) you will further explore finding deltas graphically. Use a CAS to perform the following steps: $$ \begin{array}{l}{\text { a. Plot the function } y=f(x) \text { near the point } c \text { being approached. }} \\ {\text { b. Guess the value of the limit } L \text { and then evaluate the limit sym- }} \\ {\text { bolically to see if you guessed correctly. }} \\ {\text { c. Using the value } \epsilon=0.2, \text { graph the banding lines } y_{1}=L-\epsilon} \\ {\quad \text { and } y_{2}=L+\epsilon \text { together with the function } f \text { near } c .}\end{array} $$ $$ \begin{array}{c}{\text { d. From your graph in part (c), estimate a } \delta>0 \text { such that for all } x} \\ {0<|x-c|<\delta \quad \Rightarrow \quad|f(x)-L|<\epsilon}\end{array} $$ $$ \begin{array}{l}{\text { Test your estimate by plotting } f, y_{1}, \text { and } y_{2} \text { over the interval }} \\ {0<|x-c|<\delta . \text { For your viewing window use } c-2 \delta \leq} \\ {x \leq c+2 \delta \text { and } L-2 \epsilon \leq y \leq y+2 \epsilon . \text { If any function values }}\end{array} $$$$ \begin{array}{c}{\text { lie outside the interval }[L-\epsilon, L+\epsilon], \text { your choice of } \delta} \\ {\text { was too large. Try again with a smaller estimate. }} \\ {\text { e. Repeat parts (c) and (d) successively for } \epsilon=0.1,0.05, \text { and } 0.001}\end{array} $$ $$ f(x)=\frac{\sin 2 x}{3 x}, \quad c=0 $$

Step-by-Step Solution

Answer

The limit of \( f(x) = \frac{\sin 2x}{3x} \) as \( x \to 0 \) is \( \frac{2}{3} \) with \( \delta \) depending on \( \epsilon \).

Step 1: Plot the Function

Using a computer algebra system (CAS), graph the function \( f(x) = \frac{\sin 2x}{3x} \) near the point \( c = 0 \). Analyze the graph to understand how the function behaves as \( x \) approaches \( c \).

Step 2: Estimate the Limit

From the graph gathered in Step 1, attempt to guess the value that \( f(x) \) approaches as \( x \) gets closer to \( c = 0 \). Check this guess symbolically using limits. For \( f(x) = \frac{\sin 2x}{3x} \), the limit as \( x \to 0 \) is 2/3, calculated using L'Hôpital's rule or by recognizing the standard limit \( \lim_{x \to 0} \frac{\sin x}{x} = 1 \).

Step 3: Graph the Banding Lines for \( \epsilon = 0.2 \)

On the same graph as \( f(x) \), plot the horizontal lines \( y_1 = L - \epsilon = \frac{2}{3} - 0.2 \) and \( y_2 = L + \epsilon = \frac{2}{3} + 0.2 \). Observe the behavior of \( f(x) \) as it interacts with these lines.

Step 4: Estimate \( \delta \) for \( \epsilon = 0.2 \)

Observe the graph and estimate how close \( x \) must be to \( c = 0 \) (say \( \delta \)), so that \( |f(x) - L| < \epsilon \). A visual inspection might suggest a certain range. Verify by ensuring \( f(x) \) remains between \( y_1 \) and \( y_2 \) for the interval \( 0 < |x| < \delta \). Adjust \( \delta \) if necessary.

Step 5: Validate the \( \delta \) for \( \epsilon = 0.2 \)

Plot \( f(x) \), \( y_1 \), and \( y_2 \) over the calculated \( \delta \) range. Ensure the window is set from \( c-2\delta \leq x \leq c+2\delta \) and \( L-2\epsilon \leq y \leq L+2\epsilon \). Adjust \( \delta \) if any point of \( f(x) \) exits the band.

Step 6: Repeat for Smaller \( \epsilon \) Values

For each smaller \( \epsilon \) value (0.1, 0.05, 0.001), repeat steps 3-5. Plot the bands and estimate \( \delta \) ensuring \( f(x) \) remains within each corresponding band.

Key Concepts

Graphical Analysis

Graphical analysis is a crucial method in calculus that helps us visually understand the behavior of functions. When dealing with limits, it provides an intuitive way to see how a function behaves as it approaches a specific point. For example, in the given exercise, the function \( f(x) = \frac{\sin 2x}{3x} \) is graphed near the point \( c = 0 \). This visual representation allows you to observe patterns and tendencies of the function as \( x \) approaches \( c \).

Plotting the function is the first step in visually estimating a limit. By examining how close the function gets to a certain value, we can make an educated guess about the limit. Using graphical analysis helps us see if there's any deviation or noise that might affect our estimation, allowing for finer adjustments.

Ultimately, this approach provides a means of visual verification that supports our calculations and conjectures about the limit's true value.

Delta-Epsilon Definition

The Delta-Epsilon definition of a limit is a formal method in calculus used to prove that a function approaches a particular limit. This technique involves finding two values, \( \delta \) (delta) and \( \epsilon \) (epsilon), to demonstrate with certainty that a function approaches a limit \( L \) near some point \( c \).

In simple terms, \( \epsilon \) is a small positive number representing how close \( f(x) \) should be to the limit \( L \). For every \( \epsilon \) chosen, there exists a corresponding \( \delta \) such that if \( 0 < |x-c| < \delta \), then \( |f(x) - L| < \epsilon \).

This exercise demonstrates practical application of the Delta-Epsilon concept by asking you to estimate \( \delta \) given \( \epsilon = 0.2 \) and plot banding lines at \( L-\epsilon \) and \( L+\epsilon \). The goal is to see graphically whether the function remains within these bounds, thus verifying your estimated \( \delta \).

• Provides a way to validate limit properties rigorously
• Enhances accuracy through visual and algebraic checks

L'Hôpital's Rule

L'Hôpital's Rule is a powerful mathematical tool used to find limits of indeterminate forms, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The rule simplifies these complex forms by applying derivatives. If conditions are met, this rule states:\[\lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)}\]provided the limit on the right side exists.

In the provided exercise, L'Hôpital's Rule helps find the limit of \( f(x) = \frac{\sin 2x}{3x} \) as \( x \to 0 \). Initially, \( \frac{\sin 2x}{3x} \) yields \( \frac{0}{0} \), an indeterminate form. Applying L'Hôpital's Rule involves differentiating the numerator and the denominator, allowing us to calculate the limit as \( \frac{2}{3} \).

This method is extremely useful, especially in situations involving trigonometric functions, helping to quickly find limits where direct substitution fails. It's a staple in calculus for transforming tricky limit equations into solvable iterations.

Computer Algebra System (CAS)

A Computer Algebra System (CAS) is software designed to perform symbolic mathematical computations with ease and efficiency. CAS tools can graph functions, solve equations, compute limits, and much more, providing valuable assistance when tackling complex mathematical problems.

In this exercise, a CAS is used to graph the function \( f(x) = \frac{\sin 2x}{3x} \) near \( c = 0 \). With the aid of CAS, you can visualize the behavior of the function as \( x \) nears a certain point, drastically reducing time spent on manual calculations and enhancing accuracy.

CAS also assists in verifying symbolic calculations, thus ensuring that the guessed limit aligns with what's derived algebraically. By providing tools for testing various \( \epsilon \) values in a systematic fashion, it enables a deeper understanding of the Delta-Epsilon definition and other calculus concepts.

• Automates complex calculations and graphing tasks
• Improves reliability of mathematical analysis