Problem 28

Question

, use a calculator to find the indicated limit. Use a graphing calculator to plot the function near the limit point. $$ \lim _{u \rightarrow \pi / 2} \frac{2-2 \sin u}{3 u} $$

Step-by-Step Solution

Verified

Answer

The limit is 0.

1Step 1: Understanding the Limit Expression

We need to evaluate $ \lim _{u \rightarrow \pi / 2} \frac{2-2 \sin u}{3 u} $. This involves understanding the behavior of the function $ \frac{2-2 \sin u}{3 u} $ as $ u $ approaches $ \pi / 2 $. At $ u = \pi/2 $, the function might seem to have a singularity due to the denominator, which makes direct substitution unreasonable.

2Step 2: Substitution Attempt

Attempt to substitute directly into the expression: $ \sin(\pi/2) = 1 $, so the numerator becomes $ 2-2 = 0 $, and $ u = \pi/2 $ in the denominator gives $ 3(\pi/2) $. Thus, directly substituting gives us an indeterminate form of $ \frac{0}{\text{non-zero}} $, suggesting further analysis.

3Step 3: Simplification of Expression

The expression can be rewritten for deeper understanding. A possible simplification is recognizing that at $ u \approx \pi/2 $, $ \sin u \approx 1 $. Therefore, the numerator $ 2 - 2\sin u \approx 0 $ as $ u \to \pi/2 $. This commonly indicates that the function approaches 0.

4Step 4: Use of Graphing Calculator

With a graphing calculator, input the function $ y = \frac{2 - 2\sin u}{3u} $ to observe its behavior as $ u \to \pi/2 $. Analyze the graph to see the trend of the function value near $ u = \pi/2 $. You should see the graph levels off close to 0.

5Step 5: Conclusion from Graphical Observation

From the graph, observe that as $ u $ approaches $ \pi/2 $, the function $ \frac{2 - 2\sin u}{3u} $ approaches 0. This indicates that the limit is 0.

Key Concepts

Understanding Indeterminate FormsTrigonometric Limits ExplorationVisualizing with a Graphing Calculator

Understanding Indeterminate Forms

Indeterminate forms are expressions encountered in calculus where the limit cannot be directly determined. They often arise in the form of $ \frac{0}{0} $, $ \frac{\infty}{\infty} $, or other undefined expressions when we substitute the limit point in a function. These forms signify that the standard arithmetic rules do not apply, and a deeper analysis is necessary.

For instance, when you try to evaluate the limit $ \lim _{u \rightarrow \pi / 2} \frac{2-2 \sin u}{3 u} $, direct substitution results in $ \frac{0}{\text{non-zero}} $. While this can sometimes make it look like a valid expression, the numerical value can't be determined directly.
To resolve indeterminate forms, you can often apply algebraic simplifications or use L'Hopital's Rule, provided the conditions for its application are satisfied.

This concept is essential for understanding limits, as it guides when to avoid direct substitution and seek other methods like graphical analysis or simplification.

Trigonometric Limits Exploration

Trigonometric limits involve limits of functions with trigonometric components. These can be a bit trickier due to the oscillatory nature of functions like sine and cosine. A common scenario is when the function's behavior changes rapidly near the point of interest.

For our function $ \frac{2-2\sin u}{3u} $, as $ u \to \pi/2 $, the expression $ 2 - 2\sin u $ tends towards zero because $ \sin(\pi/2) = 1 $. Knowing this can simplify the analysis, as the numerator approaches zero while the denominator does not.
These types of limits might involve fundamental identities such as $ \sin^2x + \cos^2x = 1 $, which can simplify the process.

Understanding trigonometric behavior near certain points can allow students to predict and understand limit behavior more intuitively.

Visualizing with a Graphing Calculator

A graphing calculator is a powerful tool for visualizing the behavior of complex functions near certain points of interest. It allows you to plot functions and examine the trend as the variable approaches a limit.

In the case of the function $ y = \frac{2 - 2\sin u}{3u} $, inputting this into a graphing calculator and plotting near $ u = \pi/2 $ helps reveal that the function tends towards zero.
Using a graphing tool, you can focus on a specific area of the graph by zooming in on $ u = \pi/2 $, seeing how the values behave as they get arbitrarily close to the point.
Graphing calculators also enable students to explore limits by creating a visual representation, reinforcing theoretical concepts with practical examples.

This approach helps build intuition about limiting behavior and can confirm analytical findings or suggest further analysis.

Problem 28

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