Use the Squeeze Theorem to find each limit. limx →0[x(1-cos(1)/(x))]

Name: Problem 48, Chapter 1: Calculus, Early Transcendentals
Uploaded: 2021-07-30T11:33:15-08:00
Duration: 3 min 10 s
Channel: Ma. Theresa Alin
Description: Use the Squeeze Theorem to find each limit. limx →0[x(1-cos(1)/(x))]

step 1 For this problem, we need to use squeeze theorem to determine the limit of x times 1 minus cosin

Use the Squeeze Theorem to find each limit. limx...

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Key Concepts

Squeeze Theorem

The Squeeze Theorem is a method used to determine the limit of a function by 'squeezing' it between two functions whose limits are known and equal at a point. If a function is bounded above and below by functions that share the same limit as x approaches a particular value, then the function's limit at that point is the same as those limits.

Boundedness of Trigonometric Functions

Trigonometric functions, especially cosine, are inherently bounded. In particular, the cosine function always lies between -1 and 1. This property is essential when dealing with limits involving oscillatory behavior, as it allows one to set up inequalities that facilitate the use of the Squeeze Theorem.

Handling Oscillatory Functions in Limits

When a function includes an oscillatory component, such as cosine of a term that becomes unbounded, the limit can often be evaluated by considering the bounded nature of the oscillatory part. When multiplied by another function that approaches zero, the product is dominated by the behavior of the non-oscillatory part, allowing one to conclude the limit based on the squeezing of the oscillatory term.

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