Determine the following limits.


$\underset{\theta \to 0}{\lim }\frac{2+\sin \theta }{1-{\cos }^2\theta }$

Verified step by step guidance

1. Step 1: Recognize that the denominator can be simplified using a trigonometric identity. Recall that \(1 - \cos^2\theta = \sin^2\theta\).
2. Step 2: Substitute the identity into the limit expression, transforming it into \(\lim_{\theta \to 0} \frac{2 + \sin\theta}{\sin^2\theta}\).
3. Step 3: Consider using L'Hôpital's Rule, which is applicable when the limit results in an indeterminate form like \(\frac{0}{0}\).
4. Step 4: Differentiate the numerator and the denominator separately. The derivative of the numerator \(2 + \sin\theta\) is \(\cos\theta\), and the derivative of the denominator \(\sin^2\theta\) is \(2\sin\theta\cos\theta\).
5. Step 5: Apply L'Hôpital's Rule to find the limit: \(\lim_{\theta \to 0} \frac{\cos\theta}{2\sin\theta\cos\theta}\). Simplify the expression and evaluate the limit as \(\theta\) approaches 0.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the limit of a function as theta approaches 0. Understanding limits is crucial for evaluating the behavior of functions near specific points, especially when direct substitution may lead to indeterminate forms.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are periodic functions that relate angles to ratios of sides in right triangles. In this limit problem, the sine and cosine functions are evaluated at theta, which approaches 0. Familiarity with the values of these functions at key angles, particularly their behavior near 0, is essential for simplifying the limit expression.

Indeterminate Forms

Indeterminate forms occur in calculus when direct substitution into a limit results in expressions like 0/0 or ∞/∞. In this case, substituting theta = 0 into the limit expression leads to an indeterminate form, necessitating further analysis or algebraic manipulation to resolve. Recognizing and handling these forms is vital for correctly evaluating limits.