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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 2 - Limits and Continuity문제 2.4.34
2장, 문제 2.4.34

Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


llimx→0 (x −x cos x) / sin² 3x

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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.
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.
Apply the trigonometric identity 1 - cos x ≈ (x²/2) as x approaches 0. This approximation is useful for simplifying the numerator further.
Substitute the approximation into the expression: x(x²/2) / sin²(3x). This simplifies to x³/2sin²(3x).
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.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
06:11
Limits of Rational Functions: Denominator = 0

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.
추천 영상:
6:04
Introduction to Trigonometric Functions

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.
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Slope-Intercept Form
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