4. Applications of Derivatives
Differentials
Problem 4.R.63
Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 (3 sin² 2Θ) / Θ²
Verified step by step guidance1
Identify the limit to evaluate: lim_Θ→0 (3 sin²(2Θ)) / Θ². Check if it results in an indeterminate form like 0/0 or ∞/∞ as Θ approaches 0.
Since sin(2Θ) approaches 0 as Θ approaches 0, substitute Θ = 0 into the limit to confirm it results in the indeterminate form 0/0.
Apply l'Hôpital's Rule, which states that if the limit results in 0/0 or ∞/∞, you can take the derivative of the numerator and the derivative of the denominator.
Differentiate the numerator: The derivative of 3 sin²(2Θ) using the chain rule is 6 sin(2Θ) cos(2Θ) * 2 = 12 sin(2Θ) cos(2Θ). Differentiate the denominator: The derivative of Θ² is 2Θ.
Rewrite the limit using the derivatives: lim_Θ→0 (12 sin(2Θ) cos(2Θ)) / (2Θ) and simplify before substituting Θ = 0 again.
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