4. Applications of Derivatives
Differentials
Problem 4.R.65
Textbook Question
60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.
lim_Θ→0 2Θ cot 3Θ
Verified step by step guidance1
Identify the limit to evaluate: lim_Θ→0 2Θ cot(3Θ). Recognize that cot(3Θ) can be rewritten as cos(3Θ)/sin(3Θ).
Substitute the expression into the limit: lim_Θ→0 2Θ (cos(3Θ)/sin(3Θ)). This gives you lim_Θ→0 (2Θ cos(3Θ))/sin(3Θ).
Evaluate the limit directly. As Θ approaches 0, both the numerator and denominator approach 0, indicating a 0/0 indeterminate form.
Apply l'Hôpital's Rule, which states that if you have a 0/0 form, you can take the derivative of the numerator and the derivative of the denominator.
Differentiate the numerator (2Θ cos(3Θ)) and the denominator (sin(3Θ)) separately, then re-evaluate the limit after applying l'Hôpital's Rule.
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