60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed.

lim_Θ→0 2Θ cot 3Θ

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Identify the form of the limit as \( \Theta \to 0 \). The expression \( 2\Theta \cot 3\Theta \) can be rewritten as \( \frac{2\Theta}{\tan 3\Theta} \). As \( \Theta \to 0 \), both the numerator \( 2\Theta \) and the denominator \( \tan 3\Theta \) approach 0, resulting in an indeterminate form \( \frac{0}{0} \).

Since the limit is in the indeterminate form \( \frac{0}{0} \), apply l'Hôpital's Rule, which states that \( \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)} \) if the limit is indeterminate. Differentiate the numerator and the denominator separately.

Differentiate the numerator \( 2\Theta \) with respect to \( \Theta \). The derivative is \( 2 \).

Differentiate the denominator \( \tan 3\Theta \) with respect to \( \Theta \). Using the chain rule, the derivative is \( 3 \sec^2 3\Theta \).

Substitute the derivatives back into the limit: \( \lim_{\Theta \to 0} \frac{2}{3 \sec^2 3\Theta} \). Simplify the expression and evaluate the limit as \( \Theta \to 0 \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is crucial for determining continuity, derivatives, and integrals.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule simplifies the process of finding limits in complex scenarios.

Trigonometric Functions

Trigonometric functions, such as cotangent, sine, and cosine, are essential in calculus for analyzing periodic phenomena and angles. The cotangent function, specifically, is the reciprocal of the tangent function and can be expressed in terms of sine and cosine. Understanding the behavior of these functions near specific points, like 0, is crucial for evaluating limits involving trigonometric expressions.

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