Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


llimx→0 (x −x cos x) / sin² 3x

Verified step by step guidance

1. Recognize that the problem involves a limit as x approaches 0, and the expression includes trigonometric functions. The key identity given is lim(θ→0) (sin θ / θ) = 1, which will be useful in simplifying the expression.
2. Rewrite the expression (x - x cos x) / sin²(3x) by factoring out x from the numerator: x(1 - cos x) / sin²(3x). This helps in isolating the trigonometric part of the expression.
3. Apply the trigonometric identity 1 - cos x ≈ (x²/2) as x approaches 0. This approximation is useful for simplifying the numerator further.
4. Substitute the approximation into the expression: x(x²/2) / sin²(3x). This simplifies to x³/2sin²(3x).
5. Use the identity lim(θ→0) (sin θ / θ) = 1 to simplify sin²(3x). Recognize that sin²(3x) can be rewritten as (sin(3x)/3x)² * (3x)². As x approaches 0, (sin(3x)/3x) approaches 1, allowing further simplification of the limit.

Key Concepts

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 how functions behave near specific points, especially when direct substitution is not possible due to indeterminate forms or discontinuities.

Trigonometric Limits

Trigonometric limits, such as limθ→0 sin θ / θ = 1, are fundamental in calculus for evaluating limits involving trigonometric functions. This specific limit is often used to resolve indeterminate forms and is essential for simplifying expressions where sine and cosine functions are involved near zero.

Indeterminate Forms

Indeterminate forms occur when evaluating limits results in expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. Techniques such as L'Hôpital's Rule or algebraic manipulation are used to resolve these forms and find the actual limit, ensuring a proper understanding of the function's behavior.