Estimating Limits


[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let g(θ) = (sinθ) / θ.


b. Support your conclusion in part (a) by graphing g near θ₀ = 0.

Verified step by step guidance

1. Understand the problem: We need to estimate the limit of g(θ) = (sinθ) / θ as θ approaches 0. This is a classic limit problem where direct substitution results in an indeterminate form 0/0.
2. Recall the limit property: The limit of (sinθ) / θ as θ approaches 0 is a well-known limit in calculus, often used as a fundamental limit in trigonometry and calculus.
3. Graph the function: Use a graphing calculator to plot g(θ) = (sinθ) / θ. Set the window to focus on values of θ near 0, such as from -0.1 to 0.1, to observe the behavior of the function as θ approaches 0.
4. Analyze the graph: Observe the graph near θ = 0. Notice how the function behaves and approaches a particular value as θ gets closer to 0 from both the left and the right.
5. Conclude from the graph: Based on the graph, determine the value that g(θ) approaches as θ approaches 0. This visual evidence supports the theoretical limit you may have learned in class.

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. In this case, we are interested in the limit of g(θ) as θ approaches 0. Understanding limits is crucial for analyzing the continuity and behavior of functions, especially when direct substitution leads to indeterminate forms.

Sine Function

The sine function, denoted as sin(θ), is a periodic function that relates the angle θ to the ratio of the opposite side to the hypotenuse in a right triangle. In the context of the function g(θ) = (sinθ) / θ, the behavior of sin(θ) near θ = 0 is particularly important, as it helps determine the limit of g(θ) as θ approaches 0.

Graphing Functions

Graphing functions involves plotting the values of a function on a coordinate plane to visualize its behavior. For g(θ) = (sinθ) / θ, graphing near θ₀ = 0 allows us to observe how the function behaves as it approaches this point, providing insight into the limit and confirming our analytical conclusions through visual representation.