Using limθ→0 sin θ / θ = 1

Find the limits in Exercises 23–46.

limθ→0 (1 − cos θ) / sin 2θ

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First, recognize that the limit involves trigonometric functions as θ approaches 0. We will use the known limit limθ→0 sin θ / θ = 1 to help simplify the expression.

Rewrite the expression (1 − cos θ) / sin 2θ using trigonometric identities. Recall that sin 2θ = 2 sin θ cos θ, which can be useful for simplification.

Consider the Taylor series expansion for cos θ around θ = 0: cos θ ≈ 1 - θ²/2. This approximation can help simplify the numerator 1 - cos θ.

Substitute the approximation into the expression: (1 - (1 - θ²/2)) / (2 sin θ cos θ). This simplifies to θ²/2 / (2 sin θ cos θ).

Now, apply the limit limθ→0 sin θ / θ = 1 to the expression. As θ approaches 0, sin θ ≈ θ, and cos θ ≈ 1, allowing further simplification of the limit expression.

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주요 개념

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Limit of a Function

The limit of a function describes the behavior of the function as the input approaches a particular value. In calculus, understanding limits is crucial for analyzing the continuity and differentiability of functions. For the given problem, evaluating the limit as θ approaches 0 is essential to determine the behavior of the expression (1 − cos θ) / sin 2θ.

Trigonometric Limits

Trigonometric limits involve evaluating limits that include trigonometric functions like sine and cosine. A fundamental trigonometric limit is limθ→0 sin θ / θ = 1, which is often used to simplify expressions involving small angles. This concept is key in solving the given problem, as it helps in handling the trigonometric functions as θ approaches 0.

L'Hôpital's Rule

L'Hôpital's Rule is a method for finding limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, the limit can be found by differentiating the numerator and denominator separately. This rule is useful for the given problem if direct substitution leads to an indeterminate form.

Using limθ→0 sin θ / θ = 1Find the limits in Exercises 23–46.limθ→0 (1 − cos θ) / sin 2θ