Problem 73 - Thomas Calculus

Question

You will find a graphing calculator useful. Let \(g(\theta)=(\sin \theta) / \theta\) a. Make a table of the values of \(g\) at values of \(\theta\) that approach \(\theta_{0}=0\) from above and below. Then estimate lim \(_{\theta \rightarrow 0} g(\theta)\) b. Support your conclusion in part (a) by graphing \(g\) near \(\theta_{0}=0\)

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Choosing Values for θ

To create a table, we need to select values of \( \theta \) approaching \( 0 \) from both sides. Let's choose values: \(-0.1, -0.01, -0.001, 0, 0.001, 0.01, 0.1\).

Step 2: Calculate g(θ) for Each θ

Using a calculator, compute \( g(\theta) = \frac{\sin \theta}{\theta} \) for each chosen value of \( \theta \):- For \( \theta = -0.1 \), \( g(-0.1) \approx 0.9983 \)- For \( \theta = -0.01 \), \( g(-0.01) \approx 0.9999833 \)- For \( \theta = -0.001 \), \( g(-0.001) \approx 0.999999833 \)- For \( \theta = 0 \), \( g(0) \) is undefined by direct substitution- For \( \theta = 0.001 \), \( g(0.001) \approx 0.999999833 \)- For \( \theta = 0.01 \), \( g(0.01) \approx 0.9999833 \)- For \( \theta = 0.1 \), \( g(0.1) \approx 0.9983 \)

Step 3: Observe the Table Values

Look at the values computed:- As \( \theta \rightarrow 0^- \) (approaching from the left), \( g(\theta) \) approaches 1.- As \( \theta \rightarrow 0^+ \) (approaching from the right), \( g(\theta) \) approaches 1.The estimates suggest that \( \lim_{\theta \rightarrow 0} g(\theta) = 1 \).

Step 4: Graph g(θ) Near θ=0

Use a graphing calculator to plot the function \( g(\theta) = \frac{\sin \theta}{\theta} \) over an interval close to zero, such as \([-0.1, 0.1]\).Observe that the plot appears to have a hole or undefined point at \( \theta = 0 \), yet the graph approaches the value 1 from both sides, supporting the conclusion that \( \lim_{\theta \rightarrow 0} g(\theta) = 1 \).

Step 5: Finalize the Conclusion

Based on table values and graph plot, it is highly evident that \( g(\theta) \) approaches 1 as \( \theta \) approaches 0. Thus, we conclude that \( \lim_{\theta \rightarrow 0} \frac{\sin \theta}{\theta} = 1 \).

Key Concepts

Understanding Trigonometric Functions

Trigonometric functions play a vital role in understanding angles and their relationships in various fields of mathematics and physics. They involve the ratios of the sides of a right triangle and include primary functions such as sine, cosine, and tangent.

In our exercise, we focus on the sine function. Understanding \[ g(\theta) = \frac{\sin \theta}{\theta} \] involves grasping how the sine of an angle (given in radians) relates to the angle itself, especially as the angle approaches zero.

It's important to note that as \( \theta \) approaches 0, \( \sin \theta \) approaches \( \theta \), which is the reason why we use this function to explore limits. This phenomenon is a crucial aspect when dealing with limits and continuity in calculus, providing insights into how functions behave at extremely small angles.

The Role of Graphing Calculators

When solving problems involving trigonometric limits like our exercise, graphing calculators become indispensable. They provide a visual representation of mathematical behavior, showing how functions act near specific points.

In this exercise, plotting \[ g(\theta) = \frac{\sin \theta}{\theta} \] using a graphing calculator lets us observe the function approaching the value of 1 as \( \theta \) approaches 0. This graphic representation supports the theoretical computation, making abstract concepts more tangible.

• Graphing calculators can plot multiple points quickly, saving time and reducing errors compared to manual calculations.
• They help identify limits by displaying trends and approaching values visually, which is particularly useful when the function is undefined at a point, like \( g(0) \).

By employing graphing calculators, students can explore beyond pure computation, gaining a more comprehensive understanding of mathematical phenomena.

Approaching Values and Limits

The concept of approaching values is the essence of understanding limits in calculus. It is not about reaching the exact value, but about approaching it from both sides. In our exercise, we explore this by examining how \( g(\theta) = \frac{\sin \theta}{\theta} \) behaves as \( \theta \) moves closer to 0.

Limits explain how a function's value can get infinitely close to a particular number without actually being that number. This is evident when observing that as \( \theta \) approaches 0, both from the negative and the positive side, \( g(\theta) \) approaches 1.

• From \( \theta^- \) (the left), and \( \theta^+ \) (the right), \( g(\theta) \) nears 1, showing consistency of limits from both directions.
• This consistency is vital, as it confirms the limit exists and reinforces our conclusion: \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \).

Understanding this aspect aids in developing deeper insights into continuous functions and prepares you for more complex calculus problems.