Textbook Question
Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
lim x→0 4 / x²/⁵
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1. Limits and Continuity
Introduction to Limits
1:26 minutesProblem 2.2.73b
Textbook Question
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 guidance1
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.
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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.
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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.
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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.
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