Problem 61 - Thomas Calculus

Question

In Exercises \(61-66,\) you will further explore finding deltas graphically. Use a CAS to perform the following steps: $$ \begin{array}{l}{\text { a. Plot the function } y=f(x) \text { near the point } c \text { being approached. }} \\ {\text { b. Guess the value of the limit } L \text { and then evaluate the limit sym- }} \\ {\text { bolically to see if you guessed correctly. }} \\ {\text { c. Using the value } \epsilon=0.2, \text { graph the banding lines } y_{1}=L-\epsilon} \\ {\quad \text { and } y_{2}=L+\epsilon \text { together with the function } f \text { near } c .}\end{array} $$ $$ \begin{array}{c}{\text { d. From your graph in part (c), estimate a } \delta>0 \text { such that for all } x} \\ {0<|x-c|<\delta \quad \Rightarrow \quad|f(x)-L|<\epsilon}\end{array} $$ $$ \begin{array}{l}{\text { Test your estimate by plotting } f, y_{1}, \text { and } y_{2} \text { over the interval }} \\ {0<|x-c|<\delta . \text { For your viewing window use } c-2 \delta \leq} \\ {x \leq c+2 \delta \text { and } L-2 \epsilon \leq y \leq y+2 \epsilon . \text { If any function values }}\end{array} $$$$ \begin{array}{c}{\text { lie outside the interval }[L-\epsilon, L+\epsilon], \text { your choice of } \delta} \\ {\text { was too large. Try again with a smaller estimate. }} \\ {\text { e. Repeat parts (c) and (d) successively for } \epsilon=0.1,0.05, \text { and } 0.001}\end{array} $$ $$ f(x)=\frac{x^{4}-81}{x-3}, \quad c=3 $$

Step-by-Step Solution

Answer

Limit is 108 as \( x \to 3 \). Plot and adjust \( \delta \) for correct \( \epsilon \) bands.

Step 1: Plot the Function Graph

First, plot the function \( f(x) = \frac{x^4 - 81}{x - 3} \) near \( x = 3 \) using a Computer Algebra System (CAS). Observe the behavior of the graph as \( x \) approaches 3 to help with estimating the limit.

Step 2: Estimate and Evaluate the Limit

Based on the plot, guess the value of the limit \( L \) as \( x \to 3 \). Then, compute the limit algebraically: \[L = \lim_{{x \to 3}} \frac{x^4 - 81}{x - 3} \]This can be simplified by factoring the numerator: \[= \lim_{{x \to 3}} \frac{(x^2 + 3)(x^2 - 3)}{x-3} = \lim_{{x \to 3}} (x^2 + 3)(x - 3)\]Apply L'Hôpital's Rule or further factor until you reach: \[= \lim_{{x \to 3}}(x^3 + 3x^2 + 9x + 27) = 108\]So, \( L = 108 \).

Step 3: Graph Banding Lines for Epsilon = 0.2

With \( \epsilon = 0.2 \), graph the banding lines \( y_1 = L - \epsilon = 108 - 0.2 = 107.8 \) and \( y_2 = L + \epsilon = 108 + 0.2 = 108.2 \) along with \( f(x) \) near \( x = 3 \). Check where \( f(x) \) crosses these lines.

Step 4: Estimate Delta

From the graph, determine the largest \( \delta > 0 \) for which \( 0 < |x - 3| < \delta \) ensures \( |f(x) - 108| < 0.2 \). Use trial and error by visual inspection to find \( \delta \) within which \( f(x) \) stays between the banding lines.

Step 5: Test Delta by Plotting

Test your estimated \( \delta \) by plotting \( f(x) \), \( y_1 \), and \( y_2 \) over the interval \( 3 - \delta < x < 3 + \delta \) and within the window \( L - 2\epsilon < y < L + 2\epsilon \). Adjust \( \delta \) if points lie outside the banding lines.

Step 6: Repeat for Smaller Epsiilons

Repeat Steps 3 to 5 with \( \epsilon = 0.1, 0.05, \text{ and } 0.001 \). Each time, select appropriate \( \delta \) values that satisfy the condition \( |f(x) - L| < \epsilon \) using the observation method with your CAS.

Key Concepts

Delta-epsilon definition

In calculus, the delta-epsilon definition of a limit provides a precise way to understand the concept of limits. It essentially states that a function \( f(x) \) approaches a limit \( L \) as \( x \) approaches \( c \), if for every positive number \( \epsilon \), there exists a corresponding positive number \( \delta \) such that whenever \( 0 < |x-c| < \delta \), it follows that \( |f(x) - L| < \epsilon \). This can be thought of as a game between two people: one chooses an \( \epsilon \), and the other must find a suitable \( \delta \) to keep the function within the \( \epsilon \) band around \( L \).
Think of \( \epsilon \) as the vertical tolerance on the y-axis and \( \delta \) as the horizontal tolerance around the approach point \( c \) on the x-axis. By setting \( \epsilon \) smaller and smaller, you test whether you can always find a \( \delta \) that places the function's value within the desired band around \( L \).
In practical terms, as seen in the exercise above, you'll often rely on graphing to visually determine \( \delta \) values by observing where \( f(x) \) stays within \( L \pm \epsilon \). This application is particularly useful when determining limits graphically.

Graphical limit estimation

Graphical limit estimation involves observing a function's behavior near a certain point using a graph. It's a more intuitive approach for understanding limits and can aid in making conjectures about the limit \( L \) a function approaches.
In the exercise, you start by plotting \( f(x) \) near \( x = c \) and visually identify the horizontal line to which \( f(x) \) stays closest as \( x \) nears \( c \). From the graph, you guess an approximate \( L \).
The graphical method allows insights such as:

• If the graph appears to stabilize at a certain y-value near \( c \), that is your guessed \( L \).
• By marking banding lines \( y_1 \) and \( y_2 \) (e.g., \( L \pm 0.2 \)), you see concretely how \( f(x) \) behaves within specified \( \epsilon \) bands.
• The visual tool aids in estimating a \( \delta \), giving the largest \( x \) interval around \( c \) where the function stays within the \( \epsilon \) bands.

Graphical estimation does require care, as visual inspection can sometimes be misleading, particularly when plots have limited resolution or when functions exhibit rapid oscillations.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool for evaluating limits, especially when you encounter indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The rule states that if these forms occur, the limit of the fraction \( \frac{f(x)}{g(x)} \) can be found by differentiating the numerator and the denominator separately, thus \( \lim_{{x \to c}} \frac{f(x)}{g(x)} = \lim_{{x \to c}} \frac{f'(x)}{g'(x)} \), provided the limit on the right side exists.
In the case of \( f(x) = \frac{x^4 - 81}{x - 3} \), directly substituting \( x = 3 \) gives an indeterminate form. You first try factoring the numerator to simplify. If simplification directly isn't viable, L'Hôpital's Rule can handle the form.
Here's a step-by-step breakdown when to use L’Hôpital’s Rule:

• Check whether substituting directly results in an indeterminate form.
• If so, ensure both \( f(x) \) and \( g(x) \) are differentiable near \( x = c \).
• Differentiate \( f(x) \) and \( g(x) \) separately to form \( \frac{f'(x)}{g'(x)} \).
• Evaluate the limit of the new function.

L'Hôpital's Rule is especially useful when direct algebraic simplification is cumbersome or not readily apparent.