Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limx→0 (1 − cos 3x) / 2x

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Recognize that the limit involves a trigonometric function and can be approached using trigonometric identities and the known limit lim(θ→0) (sin θ / θ) = 1.

Rewrite the expression (1 - cos 3x) using the trigonometric identity: 1 - cos A = 2sin²(A/2). In this case, A = 3x, so 1 - cos 3x = 2sin²(3x/2).

Substitute the identity into the limit expression: lim(x→0) (2sin²(3x/2) / 2x). This simplifies to lim(x→0) (sin²(3x/2) / x).

To further simplify, use the substitution u = 3x/2, which implies that as x approaches 0, u also approaches 0. Therefore, x = (2/3)u, and the limit becomes lim(u→0) (sin²(u) / ((2/3)u)).

Apply the limit property lim(u→0) (sin u / u) = 1 to the expression: lim(u→0) (sin²(u) / u) = (lim(u→0) (sin u / u)) * (lim(u→0) sin u) = 1 * 0 = 0. Thus, the original limit evaluates to 0.

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Key Concepts

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

Limit Definition

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, particularly where they may not be explicitly defined. The notation lim x→a f(x) indicates the limit of f(x) as x approaches a.

Trigonometric Limits

Trigonometric limits, such as lim θ→0 (sin θ / θ) = 1, are essential in calculus for evaluating limits involving trigonometric functions. This specific limit is crucial for deriving derivatives of sine and cosine functions and is often used in conjunction with L'Hôpital's Rule or Taylor series expansions to simplify complex limit problems.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating 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 separately. This rule simplifies the process of finding limits that are otherwise difficult to evaluate directly.

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