Textbook Question
Finding Limits
In Exercises 25–28, find the limit of g(x) as x approaches the indicated value.
lim (4g(x))¹/³ = 2
x →0
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1. Limits and Continuity
Finding Limits Algebraically
2:05 minutesProblem 2.43
Textbook Question
Limits and Infinity
Find the limits in Exercises 37–46.
sin x
lim ------------- ( If you have a grapher, try graphing
x→∞ |x| the function for ―5 ≤ x ≤ 5 ) .
Verified step by step guidance1
Identify the function for which you need to find the limit: \( \frac{\sin x}{|x|} \).
Understand that as \( x \to \infty \), the absolute value \( |x| \) behaves like \( x \) because \( x \) is positive.
Recognize that the sine function, \( \sin x \), oscillates between -1 and 1 for all real numbers \( x \).
Consider the behavior of the fraction \( \frac{\sin x}{x} \) as \( x \to \infty \). Since \( \sin x \) is bounded and \( x \) grows without bound, the fraction approaches zero.
Conclude that the limit of \( \frac{\sin x}{|x|} \) as \( x \to \infty \) is 0, because the numerator is bounded while the denominator increases indefinitely.
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Video duration:
2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions at specific points, including points of discontinuity or infinity. In this context, evaluating the limit as x approaches infinity allows us to analyze the long-term behavior of the function sin(x)/|x|.
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Behavior of Functions at Infinity
When analyzing limits as x approaches infinity, we assess how a function behaves as its input grows without bound. This often involves determining whether the function approaches a finite value, diverges to infinity, or oscillates. For the function sin(x)/|x|, understanding its behavior as x becomes very large is crucial for finding the limit.
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Trigonometric Functions
Trigonometric functions, such as sine, exhibit periodic behavior, oscillating between fixed values. The function sin(x) oscillates between -1 and 1, which is important when considering its limit in conjunction with |x|. This periodic nature influences the overall limit of the function sin(x)/|x| as x approaches infinity, as the oscillation will be divided by an increasingly large denominator.
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