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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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.

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