Using limθ→0 sin θ / θ = 1


Find the limits in Exercises 23–46.


limθ→0 tan θ / θ²cot 3θ

Verified step by step guidance

1. First, recognize that the limit involves trigonometric functions and their behavior as θ approaches 0. We will use the known limit limθ→0 sin θ / θ = 1 to help simplify the expression.
2. Rewrite tan θ in terms of sin θ and cos θ: tan θ = sin θ / cos θ. Similarly, rewrite cot 3θ as cos 3θ / sin 3θ.
3. Substitute these expressions into the limit: limθ→0 (sin θ / cos θ) / (θ² * (cos 3θ / sin 3θ)).
4. Simplify the expression: limθ→0 (sin θ * sin 3θ) / (θ² * cos θ * cos 3θ).
5. Apply the limit properties and the known limit limθ→0 sin θ / θ = 1 to evaluate the limit as θ approaches 0. Consider the behavior of each trigonometric function and their derivatives at θ = 0.

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 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 tan θ / θ²cot 3θ.

Trigonometric Limits

Trigonometric limits involve evaluating limits that include trigonometric functions like sine, cosine, and tangent. A fundamental trigonometric limit is limθ→0 sin θ / θ = 1, which is often used to simplify expressions involving small angles. This concept is key in solving the given problem, as it helps in simplifying the trigonometric components of the expression.

L'Hôpital's Rule

L'Hôpital's Rule is a method for finding limits of indeterminate forms like 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) as x approaches a value results in an indeterminate form, the limit can be found by differentiating the numerator and denominator separately. This rule is useful in the given problem if direct substitution leads to an indeterminate form, allowing for further simplification.