Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 θcos θ

Verified step by step guidance

1. Recognize that the limit involves a trigonometric function and a variable approaching zero. The expression given is \( \lim_{\theta \to 0} \theta \cos \theta \).
2. Understand that \( \cos \theta \) is a continuous function, and as \( \theta \to 0 \), \( \cos \theta \to \cos(0) = 1 \).
3. Rewrite the limit expression by separating the terms: \( \lim_{\theta \to 0} \theta \cos \theta = \lim_{\theta \to 0} \theta \cdot \lim_{\theta \to 0} \cos \theta \).
4. Substitute the known limit of \( \cos \theta \) as \( \theta \to 0 \), which is 1, into the expression: \( \lim_{\theta \to 0} \theta \cdot 1 \).
5. Evaluate the limit of \( \theta \) as \( \theta \to 0 \), which is 0. Therefore, the entire expression evaluates to \( 0 \cdot 1 = 0 \).

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, understanding limits is crucial for analyzing the behavior of functions near specific points, especially when direct substitution may lead to indeterminate forms.

Trigonometric Limits

Trigonometric limits often involve functions like sine and cosine, particularly as the angle approaches zero. A key result is that limθ→0 sin θ / θ = 1, which is foundational for evaluating limits involving trigonometric functions and is frequently used in calculus to simplify expressions.

Continuity and Differentiability

A function is continuous at a point if the limit as the input approaches that point equals the function's value at that point. Differentiability, which requires continuity, is essential for understanding how functions behave and change, particularly when evaluating limits that involve products of functions, such as θcos θ.