Skip to main content
Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.2.73b
Chapter 2, Problem 2.2.73b

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.
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.
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.
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.
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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.
Recommended video:
05:50
One-Sided Limits

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.
Recommended video:
Guided course
5:53
Graph of Sine and Cosine Function

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.
Recommended video:
Guided course
5:53
Graph of Sine and Cosine Function
Related Practice
Textbook Question

Limits and Continuity


On what intervals are the following functions continuous?


b. g(x) = csc x

1
views
Textbook Question

Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


b. lim x→0⁻ 2 / (3x¹/³)

Textbook Question

Estimating Limits


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


Let f(x)=(x² − 1)/(|x| − 1).


b. Support your conclusion in part (a) by graphing f near c = -1 and using Zoom and Trace to estimate y-values on the graph as x→−1.

Textbook Question

Estimating Limits


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


Let f(x) = (x² - 9) / (x + 3)


b. Support your conclusions in part (a) by graphing f near c = -3 and using Zoom and Trace to estimate y-values on the graph as x → −3.

Textbook Question

Estimating Limits


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


Let h(x)=(x² − 2x − 3)/(x² − 4x + 3)


b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as x→3.

Textbook Question

Estimating Limits


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


Let G(x)=(x + 6)/(x² + 4x − 12)


b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x→−6.