Problem 65 - Thomas Calculus

Question

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

Step-by-Step Solution

Answer

The limit of the function as \( x \to 1 \) is \( \frac{1}{3} \). Adjust \( \delta \) to confine \( f(x) \) in \( [L-\epsilon, L+\epsilon] \) for various \( \epsilon \) values.

Step 1: Plot the Function

First, plot the function \( f(x) = \frac{\sqrt[3]{x} - 1}{x - 1} \) near \( x = 1 \). Use a Computer Algebra System (CAS) to do this and focus on the interval around \( x = 1 \), such as \( x = 0.5 \) to \( x = 1.5 \). This helps visualize the behavior of the function near the point of interest.

Step 2: Guess and Evaluate the Limit

By observing the graph near \( x = 1 \), guess the value of the limit \( L \). Next, symbolically evaluate the limit using the CAS: \( \lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{x - 1} \). You will find this limit to be \( \frac{1}{3} \). Confirm your guess by checking this calculated value.

Step 3: Graphing with Epsilon Band

For \( \epsilon = 0.2 \), draw the lines \( y_1 = L - \epsilon = \frac{1}{3} - 0.2 \) and \( y_2 = L + \epsilon = \frac{1}{3} + 0.2 \). Graph these lines together with \( f(x) \) centered around \( x = 1 \). You are visualizing the region within which \( f(x) \) should stay.

Step 4: Estimate Delta

From the graph obtained in Step 3, estimate a \( \delta > 0 \) such that whenever \( 0 < |x - 1| < \delta \), it follows that \( |f(x) - \frac{1}{3}| < 0.2 \). Use a trial-and-error method by adjusting \( \delta \) on your CAS to ensure no part of the curve \( f(x) \) leaves the band defined by \( y_1 \) and \( y_2 \). Suppose \( \delta = 0.1 \).

Step 5: Verify Delta Estimate

Test your \( \delta \) by plotting \( f(x) \), \( y_1 \), and \( y_2 \) over the interval \( 1 - 2\delta \leq x \leq 1 + 2\delta \) and make sure values \( f(x) \) are within \( L - 2\epsilon \leq y \leq L + 2\epsilon \). If points fall outside, reduce \( \delta \) and repeat.

Step 6: Repeat for Smaller Epsilon

Repeat Steps 3 to 5 for smaller values of \( \epsilon \): 0.1, 0.05, and 0.001. Each time, you will estimate a new \( \delta \) ensuring \( f(x) \) stays within the respective bands \( L-\epsilon \) to \( L+\epsilon \).

Key Concepts

Delta-Epsilon Definition

The delta-epsilon definition is a fundamental approach to understanding limits in calculus. This definition provides a precise way to express that a function approaches a certain limit as the input approaches a particular point. In simple terms, it is a way to show that as the input of a function gets arbitrarily close to a specific value, the function's output gets arbitrarily close to the limit.
Here's how it works:

• Let the function be denoted as \( f(x) \).
• Suppose we want to prove that \( \lim_{{x \to c}} f(x) = L \).
• Using the definition, for every \( \epsilon > 0 \), there must exist a \( \delta > 0 \) such that whenever \( 0 < |x-c| < \delta \), it implies \( |f(x) - L| < \epsilon \).

Here, \( \epsilon \) represents how close we want the function value \( f(x) \) to be to the limit \( L \), whereas \( \delta \) denotes how close \( x \) should be to \( c \). The goal is to find these values such that the above criteria hold true.
The delta-epsilon approach is a very meticulous way to confirm and validate the behavior of functions as they near particular points.

Cubic Root Functions

Cubic root functions involve expressions where the variable is under a cube root. The simplest form is \( \sqrt[3]{x} \), and these functions have specific properties that make them interesting to explore.
Some key characteristics include:

• The domain of \( \sqrt[3]{x} \) is all real numbers, meaning it can accept any real input.
• The range is also all real numbers, indicating a corresponding output for every input.
• Cubic root functions are odd, meaning they exhibit symmetry. Specifically, \( \sqrt[3]{-x} = -\sqrt[3]{x} \).

When finding limits involving cubic root functions, such as \( f(x) = \frac{\sqrt[3]{x} - 1}{x - 1} \), it's crucial to analyze the function closely, especially near points where it could be undefined, such as \( x = 1 \).
Visualizing these functions graphically helps in understanding their behavior and interaction with limits at specific points.

Graphical Limit Estimation

Graphical estimation is a practical method that aids in visually comprehending the behavior of functions nearing a particular point. It's especially beneficial when dealing with complex functions where algebraic manipulation alone might be challenging.
Here's how to estimate limits graphically:

• Plot the function around the point \( x = c \) to see how it behaves.
• Overlay horizontal baseline lines, often called epsilon bands, at \( y = L - \epsilon \) and \( y = L + \epsilon \). These lines denote the acceptable range for the function's value based on the specified \( \epsilon \).
• Examining the plot, identify a \( \delta \) such that within the interval \( 0 < |x-c| < \delta \), the function remains between these horizontal lines.

This technique ties nicely with the delta-epsilon definition, as it visually represents the conditions needed for the limit to exist.
Graphical estimation allows students to interact with the function intuitively and refine their understanding of limits through observation and approximation. By adjusting \( \delta \) and \( \epsilon \), one can ensure the function's behavior aligns with the theoretical delta-epsilon criteria. This practice solidifies foundational concepts in calculus and builds problem-solving confidence.