Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.5.82

Chapter 3, Problem 3.5.82

Another method for proving lim x→0 cos x−1/x = 0 Use the half-angle formula sin²x = 1− cos 2x/2 to prove that lim x→0 cos x−1/x=0.

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Start by recognizing that the limit we want to prove is lim x→0 (cos x - 1)/x = 0. This can be challenging directly, so we use trigonometric identities to simplify the expression.

Recall the half-angle identity: sin²x = (1 - cos 2x)/2. This identity can help us relate sine and cosine functions, which is useful for limits involving trigonometric functions.

To use this identity, consider the expression for sin²(x/2) = (1 - cos x)/2. This is derived by substituting x with x/2 in the half-angle formula.

Rearrange the expression to solve for cos x: cos x = 1 - 2sin²(x/2). This expression allows us to express cos x in terms of sin(x/2), which is useful for limits as x approaches 0.

Substitute cos x = 1 - 2sin²(x/2) into the original limit expression: (cos x - 1)/x = (1 - 2sin²(x/2) - 1)/x = -2sin²(x/2)/x. Now, analyze the behavior of this expression as x approaches 0, using the fact that sin(x/2) approaches 0 as x approaches 0.

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Key Concepts

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

Limits

In calculus, a limit is a fundamental concept 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, particularly where they may not be explicitly defined. For example, evaluating the limit of a function as x approaches 0 can reveal insights about continuity and differentiability.

Trigonometric Identities

Trigonometric identities are equations that involve trigonometric functions and are true for all values of the variables involved. The half-angle formula, sin²x = (1 - cos(2x))/2, is a specific identity that relates the sine and cosine functions. These identities are essential for simplifying expressions and solving limits involving trigonometric functions.

L'Hôpital's Rule

L'Hôpital's Rule is a method in calculus used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when direct substitution in limit problems leads to undefined expressions.

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