Find the limits in Exercises 49–52. Write ∞ or −∞ where appropriate.


lim θ→0 (2 − cot θ)

Verified step by step guidance

1. First, recall the definition of cotangent: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). This will help us rewrite the expression in terms of sine and cosine.
2. Substitute \( \cot \theta \) with \( \frac{\cos \theta}{\sin \theta} \) in the limit expression: \( \lim_{\theta \to 0} (2 - \frac{\cos \theta}{\sin \theta}) \).
3. To simplify the expression, find a common denominator: \( \lim_{\theta \to 0} \left( \frac{2\sin \theta - \cos \theta}{\sin \theta} \right) \).
4. Evaluate the limit by considering the behavior of \( \sin \theta \) and \( \cos \theta \) as \( \theta \to 0 \). Note that \( \sin \theta \to 0 \) and \( \cos \theta \to 1 \).
5. Apply L'Hôpital's Rule if necessary, which is used when the limit results in an indeterminate form like \( \frac{0}{0} \). Differentiate the numerator and the denominator separately and then evaluate the limit again.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They help in understanding the behavior of functions near specific points, including points of discontinuity or infinity. Evaluating limits is crucial for defining derivatives and integrals, which are core concepts in calculus.

Cotangent Function

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function, defined as cot(θ) = cos(θ)/sin(θ). It is important to understand its behavior, especially near critical points like θ = 0, where it approaches infinity. This function plays a significant role in trigonometric limits and can affect the outcome of limit evaluations.

Indeterminate Forms

Indeterminate forms occur in limit problems when direct substitution leads to ambiguous results, such as 0/0 or ∞ - ∞. Recognizing these forms is essential for applying techniques like L'Hôpital's Rule or algebraic manipulation to resolve the limit. Understanding how to handle indeterminate forms is key to successfully finding limits in calculus.