Textbook Question
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 sin θ / sin 2θ
1. Limits and Continuity
Introduction to Limits
4:05 minutesProblem 2.4.2g
Textbook Question
Finding Limits Graphically
Which of the following statements about the function y = f(x) graphed here are true, and which are false?
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g. limx→0+ f(x) = limx→0− f(x)
Verified step by step guidance1
To determine if the statement limx→0+ f(x) = limx→0− f(x) is true, we need to analyze the graph of the function y = f(x) as x approaches 0 from both the right (0+) and the left (0−).
Examine the behavior of the function as x approaches 0 from the right (x → 0+). Look at the values of f(x) as x gets closer to 0 from positive values. This is the right-hand limit.
Next, examine the behavior of the function as x approaches 0 from the left (x → 0−). Look at the values of f(x) as x gets closer to 0 from negative values. This is the left-hand limit.
Compare the right-hand limit and the left-hand limit. If both limits are equal, then limx→0+ f(x) = limx→0− f(x) is true. If they are not equal, the statement is false.
Conclude whether the statement is true or false based on the comparison of the right-hand and left-hand limits. This will tell you if the function has a two-sided limit at x = 0.
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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. It helps in understanding how functions behave near points of interest, including points of discontinuity. The notation limx→a f(x) indicates the value that f(x) approaches as x approaches a from either the left (−) or the right (+).
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One-Sided Limits
One-Sided Limits
One-sided limits refer to the limits of a function as the input approaches a specific value from one side only. The right-hand limit, denoted as limx→a+ f(x), considers values approaching 'a' from the right, while the left-hand limit, limx→a− f(x), considers values approaching from the left. For a limit to exist at a point, both one-sided limits must be equal.
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Continuity
A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. This means there are no breaks, jumps, or holes in the graph at that point. Understanding continuity is crucial for determining the validity of statements regarding limits, especially when evaluating whether the left-hand and right-hand limits are equal.
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