Problem 14 - Calculus: Early Transcendentals

Question

Finding \(\delta\) for a given \(\varepsilon\) using a graph Let \(g(x)=2 x^{3}-12 x^{2}+26 x+4\) and note that \(\lim _{x \rightarrow 2} g(x)=24\) For each value of \(\varepsilon\), use a graphing utility to find all values of \(\delta>0\) such that \(|g(x)-24|<\varepsilon\) whenever \(0<|x-2|<\delta\) Sketch graphs illustrating your work. a. \(\varepsilon=1\) b. \(\varepsilon=0.5\)

Step-by-Step Solution

Answer

Question: For the given limit theorem, \(\lim _{x \rightarrow 2} g(x)=24\), find the values of \(\delta\) for \(\varepsilon=1\) and \(\varepsilon=0.5\), and describe how you used graphs to find these values.

Step 1: Graph \(g(x)\) and \(y=24\pm1\)

Using your graphing utility, graph the function \(g(x)=2x^3-12x^2+26x+4\) and the horizontal lines \(y=23\) and \(y=25\). These lines represent the bounds of the given \(\varepsilon=1\). 2. Find the intersection points

Step 2: Identify the intersection points

Locate where the graph of \(g(x)\) intersects the lines \(y=23\) and \(y=25\). These intersections represent the points where \(|g(x)-24|=1\). 3. Calculate the values of \(\delta\)

Step 3: Calculate values of \(\delta\)

Read the \(x\)-coordinates of the intersection points found in the previous step and calculate the distance from these points to the \(x\)-value of \(2\). The smallest absolute distance to these intersections on each side will determine the values of \(\delta\) that satisfy the theorem for \(\varepsilon=1\). b. Finding values of \(\delta\) for \(\varepsilon=0.5\) 4. Graphing the function with new bounds

Step 4: Graph \(g(x)\) and \(y=24\pm0.5\)

Using your graphing utility, graph the function \(g(x)=2x^3-12x^2+26x+4\) and the horizontal lines \(y=23.5\) and \(y=24.5\). These lines represent the bounds of the given \(\varepsilon=0.5\). 5. Find the intersection points for the new bounds

Step 5: Identify the intersection points

Locate where the graph of \(g(x)\) intersects the lines \(y=23.5\) and \(y=24.5.\) These intersections represent the points where \(|g(x)-24|=0.5\). 6. Calculate the values of \(\delta\) for the new bounds

Step 6: Calculate values of \(\delta\)

Read the \(x\)-coordinates of the intersection points found in step 5 and calculate the distance from these points to the \(x\)-value of \(2\). The smallest absolute distance to these intersections on each side will determine the values of \(\delta\) that satisfy the theorem for \(\varepsilon=0.5\). Throughout this process, we have found the values of \(\delta\) for the given values of \(\varepsilon\) and have sketched graphs to illustrate the results.

Key Concepts

Graphical Analysis of Limits

Understanding limits through graphical analysis can be very intuitive. With visuals, you can better comprehend how functions behave as they approach specific points. For the function \( g(x) = 2x^3 - 12x^2 + 26x + 4 \), graphing is particularly useful for finding \( \delta \) given a specific \( \varepsilon \).

• Begin by plotting the curve of \( g(x) \). Notice its shape as it fluctuates around a horizontal line.
• Draw lines at \( y = 24 \pm \varepsilon \). These lines set your desired margin of difference from 24.
• Track where the curve intersects these horizontal lines to determine the range where the function is constrained within \( \varepsilon \).

Graphical analysis is helpful because it provides a visual way to identify values of \( \delta \) that satisfy the condition \( |g(x) - 24| < \varepsilon \).

Delta-Epsilon Definition of Limits

The delta-epsilon definition is a cornerstone in understanding limits. It provides a precise mathematical way to express the idea that a function gets close to a limit. This concept is crucial for grasping continuity and the behavior of functions.

The definition states:

• For every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |x - a| < \delta \), then \( |f(x) - L| < \varepsilon \).

In our exercise, we used this definition to find specific \( \delta \) values for given \( \varepsilon \) values, effectively showing how \( g(x) \) approaches 24 as \( x \) approaches 2.

Visualizing this with graphs, as we did, shows the intervals that correspond to the calculated \( \delta \) values. This solidifies understanding by relating numeric comprehension with spatial visualization.

Understanding Polynomial Functions

Polynomial functions like \( g(x) = 2x^3 - 12x^2 + 26x + 4 \) consist of several terms with varying powers of \( x \). Such functions are smooth and continuous, hence often used in calculus for study and analysis.

Key characteristics of polynomial functions:

• They can be easily manipulated algebraically.
• Graphs of polynomials are continuous and smooth, curves without breaks or sharp bends.
• The degree of the polynomial (largest exponent) hints at the number of intersecting points the curve might have with a horizontal line.

Understanding these properties helps when determining limits, as polynomial functions do not have any strange discontinuities or asymptotic behavior to consider.

Understanding Continuity of Functions

Continuity is a fundamental property where functions have no breaks, jumps, or holes. For the function \( g(x) \), which is a polynomial, we assume it is continuous everywhere.

What does continuity entail?

• A function \( f(x) \) is continuous at a point \( a \) if \( \lim_{x \to a} f(x) = f(a) \).
• For any chosen \( \varepsilon > 0 \), you can find a \( \delta > 0 \) fulfilling the \( |f(x) - f(a)| < \varepsilon \) condition whenever \( 0 < |x-a| < \delta \).
• Polynomials like \( g(x) \) are inherently continuous, simplifying calculation of limits using delta-epsilon definitions.

Graphically, this means when you zoom in on any part of the graph of a polynomial, it will consistently be a smooth curve, reaffirming its continuous nature.