Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 sin θ / sin 2θ

Verified step by step guidance

1. Recognize that the limit involves the expression \( \frac{\sin \theta}{\sin 2\theta} \) as \( \theta \to 0 \). We need to manipulate this expression to use the known limit \( \lim_{\theta \to 0} \frac{\sin \theta}{\theta} = 1 \).
2. Rewrite \( \sin 2\theta \) using the double angle identity: \( \sin 2\theta = 2\sin \theta \cos \theta \). This gives us \( \frac{\sin \theta}{\sin 2\theta} = \frac{\sin \theta}{2\sin \theta \cos \theta} = \frac{1}{2\cos \theta} \).
3. Now, consider the limit \( \lim_{\theta \to 0} \frac{1}{2\cos \theta} \). Since \( \cos \theta \to 1 \) as \( \theta \to 0 \), substitute this into the expression.
4. Evaluate the limit: \( \lim_{\theta \to 0} \frac{1}{2\cos \theta} = \frac{1}{2 \cdot 1} \).
5. Conclude that the limit is \( \frac{1}{2} \).

Key Concepts

Limit of a Function

The limit of a function describes the value that a function approaches as the input approaches a certain point. In calculus, limits are fundamental for understanding continuity, derivatives, and integrals. The notation limθ→0 indicates that we are examining the behavior of the function as θ approaches 0.

Sine Function Behavior

The sine function, sin(θ), is a periodic function that oscillates between -1 and 1. As θ approaches 0, sin(θ) behaves similarly to its argument, meaning sin(θ) approaches θ. This property is crucial for evaluating limits involving sine, particularly in the context of small angles.

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(θ)/g(θ) results in an indeterminate form, the limit can be evaluated by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful when direct substitution does not yield a clear result.