Determine the following limits.
lim θ→π/2 sin^2 θ − 5 sin θ + 4 / sin^2 θ − 1

Verified step by step guidance

1. Identify the limit expression: \( \lim_{\theta \to \frac{\pi}{2}} \frac{\sin^2 \theta - 5 \sin \theta + 4}{\sin^2 \theta - 1} \).
2. Substitute \( \theta = \frac{\pi}{2} \) into the expression to check for indeterminate form. Since \( \sin\left(\frac{\pi}{2}\right) = 1 \), the expression becomes \( \frac{1 - 5 \cdot 1 + 4}{1 - 1} = \frac{0}{0} \), which is indeterminate.
3. Factor the numerator \( \sin^2 \theta - 5 \sin \theta + 4 \) as \( (\sin \theta - 4)(\sin \theta - 1) \).
4. Factor the denominator \( \sin^2 \theta - 1 \) as \( (\sin \theta - 1)(\sin \theta + 1) \).
5. Cancel the common factor \( (\sin \theta - 1) \) from the numerator and the denominator, then evaluate the limit of the simplified expression as \( \theta \to \frac{\pi}{2} \).

Key Concepts

Limits

In calculus, a limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points where the function may not be explicitly defined. Evaluating limits is essential for determining continuity, derivatives, and integrals.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate angles to ratios of sides in right triangles. In this limit problem, the sine function is used, and understanding its properties, such as periodicity and specific values at key angles (like π/2), is crucial for evaluating the limit accurately. These functions often appear in calculus problems involving limits and derivatives.

Factoring and Simplifying Expressions

Factoring and simplifying expressions is a key algebraic skill that aids in evaluating limits, especially when direct substitution leads to indeterminate forms like 0/0. In this limit, simplifying the expression by factoring the numerator and denominator can reveal the limit's value as θ approaches π/2, making it easier to compute the limit without encountering undefined behavior.