Problem 62 - Thomas Calculus

Question

In Exercises \(61-66,\) you will further explore finding deltas graphically. Use a CAS to perform the following steps: $$ \begin{array}{l}{\text { a. Plot the function } y=f(x) \text { near the point } c \text { being approached. }} \\ {\text { b. Guess the value of the limit } L \text { and then evaluate the limit sym- }} \\ {\text { bolically to see if you guessed correctly. }} \\ {\text { c. Using the value } \epsilon=0.2, \text { graph the banding lines } y_{1}=L-\epsilon} \\ {\quad \text { and } y_{2}=L+\epsilon \text { together with the function } f \text { near } c .}\end{array} $$ $$ \begin{array}{c}{\text { d. From your graph in part (c), estimate a } \delta>0 \text { such that for all } x} \\ {0<|x-c|<\delta \quad \Rightarrow \quad|f(x)-L|<\epsilon}\end{array} $$ $$ \begin{array}{l}{\text { Test your estimate by plotting } f, y_{1}, \text { and } y_{2} \text { over the interval }} \\ {0<|x-c|<\delta . \text { For your viewing window use } c-2 \delta \leq} \\ {x \leq c+2 \delta \text { and } L-2 \epsilon \leq y \leq y+2 \epsilon . \text { If any function values }}\end{array} $$$$ \begin{array}{c}{\text { lie outside the interval }[L-\epsilon, L+\epsilon], \text { your choice of } \delta} \\ {\text { was too large. Try again with a smaller estimate. }} \\ {\text { e. Repeat parts (c) and (d) successively for } \epsilon=0.1,0.05, \text { and } 0.001}\end{array} $$ $$ f(x)=\frac{5 x^{3}+9 x^{2}}{2 x^{5}+3 x^{2}}, \quad c=0 $$

Step-by-Step Solution

Answer

Limit is 3; \( \delta \) must be small enough to ensure \( |f(x) - L| < \epsilon \).

Step 1: Plot the Function

Using a CAS, plot the function \( y = f(x) = \frac{5x^{3} + 9x^{2}}{2x^{5} + 3x^{2}} \) near \( c = 0 \). Focus on getting a clearer view around \( x = 0 \) to observe the behavior of the function.

Step 2: Guess the Limit

Analyze the plot from Step 1 to guess the limit \( L \). As \( x \) approaches \( 0 \), observe the trend in \( y = f(x) \). It appears \( L \), the limit, is about \( 3 \).

Step 3: Confirm the Limit

Evaluate the limit \( \lim_{x \to 0} f(x) \) symbolically using the CAS to determine if our guess \( L = 3 \) is correct. The symbolic computation verifies that the limit \( L \) is indeed 3.

Step 4: Graph the Banding Lines for \( \epsilon = 0.2 \)

With \( \epsilon = 0.2 \), draw the horizontal lines \( y_1 = L - \epsilon = 2.8 \) and \( y_2 = L + \epsilon = 3.2 \) on the same graph as \( f(x) \), near \( c = 0 \). This creates a band around the limit.

Step 5: Estimate \( \delta \) for \( \epsilon = 0.2 \)

Visually inspect the graph to find a \( \delta > 0 \) such that for \( 0 < |x - c| < \delta \), \( |f(x) - L| < 0.2 \). A reasonable estimate for \( \delta \) might be \( 0.5 \).

Step 6: Test \( \delta \) for \( \epsilon = 0.2 \)

Plot \( f(x) \), \( y_1 \), and \( y_2 \) within the interval \( 0 < |x - c| < \delta \), ensuring \( c - 2\delta \leq x \leq c + 2\delta \) and \( L - 2\epsilon \leq y \leq L + 2\epsilon \). If the function values stay within the band, \( \delta \) is appropriate.

Step 7: Repeat Process for Smaller \( \epsilon \) Values

Repeat Steps 4-6 for the values \( \epsilon = 0.1, 0.05, \text{ and } 0.001 \), adjusting \( \delta \) as needed each time to keep \( |f(x) - L| < \epsilon \). This will likely require decreasing \( \delta \) as \( \epsilon \) decreases.

Key Concepts

Continuity

In calculus, continuity is a fundamental concept that describes how a function behaves at a particular point. A function is said to be continuous at a point if there are no interruptions in the graph at that point. This means that as you approach a certain point on the x-axis, the values of the function smoothly approach a specific value without jumping or skipping values.
To be more precise, a function is continuous at a point \( x = c \) if the following three conditions are met:

• The function \( f(x) \) is defined at \( x = c \).
• The limit of the function \( f(x) \) as \( x \) approaches \( c \) exists.
• The limit of the function as \( x \) approaches \( c \) equals \( f(c) \).

When all of these conditions are satisfied, the function doesn't just approach a value—it becomes that value at \( x = c \). Understanding continuity helps in predicting function behavior near given points and is essential for performing various calculus operations like differentiation and integration.

Delta-Epsilon Definition

The delta-epsilon definition is a formal way to define what it means for a function to have a limit at a particular point. It's a precise mathematical method that avoids ambiguity by using specific quantities: \( \epsilon \) and \( \delta \).
In simple terms, given a function \( f(x) \), a limit \( L \), and a point \( x = c \), the goal is to show that as \( x \) gets closer to \( c \), \( f(x) \) gets closer to \( L \). Formally, we say \( \lim_{x \to c} f(x) = L \) if for every \( \epsilon > 0 \), there exists a \( \delta > 0 \) such that whenever \( 0 < |x - c| < \delta \), it follows that \( |f(x) - L| < \epsilon \).
This means:

• \( \epsilon \) represents a small positive distance around \( L \), showing how close \( f(x) \) should be to \( L \).
• \( \delta \) represents a small positive distance around \( c \), showing how close \( x \) should be to \( c \).

By adjusting \( \delta \) for any \( \epsilon \), we can demonstrate that the function approaches the limit \( L \) as closely as desired. This rigorous approach ensures consistencies in calculus and is crucial for proving limits and the continuity of functions.

Graphical Analysis

Graphical analysis involves studying and interpreting graphs to understand the behavior of functions. When dealing with limits and continuity, graphing helps visually identify how a function approaches a certain value as \( x \) nears a given point.
Using a graph, we can observe the function \( y = f(x) \) as it approaches the limit \( L \) near \( x = c \). Graphical tools allow us to:

• Identify trends or patterns in the graph of the function.
• Estimate the limit \( L \) by looking at how \( y = f(x) \) behaves as \( x \) gets closer to \( c \).
• Verify continuity by checking for any gaps or jumps in the graph.
• Draw horizontal banding lines to create a visual "band" around the estimated limit \( L \), and check if the function remains within those bounds.

Graphical analysis is particularly helpful when combined with algebraic techniques to confirm predictions about limits and continuity. The visual representation can often provide insights that are not immediately apparent through symbolic manipulation alone.

Calculus

Calculus is a branch of mathematics that deals with change and motion. It encompasses two main areas: differentiation and integration, and is instrumental in exploring limits, continuity, and the behavior of functions.
In the context of the exercise, calculus helps us understand:

• **Limits:** Calculus provides the tools to find and verify limits using both graphical and symbolic methods.
• **Derivative Analysis:** It lets us study the rate at which functions change, providing insights into their behavior at specific points.
• **Integration and Area:** While integration isn't directly part of limit finding, understanding areas under curves is a related concept.
• **Continuity and Function Behavior:** Calculus helps in establishing whether functions are continuous and describing how they behave near certain points.

By using calculus, students can develop a deeper understanding of how functions work and how to manipulate them mathematically. Mastering these concepts allows for solving complex real-world problems involving rates of change and accumulation.