Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_Θ→π/2 (tan Θ - secΘ)

Verified step by step guidance

1. First, identify the form of the limit as Θ approaches π/2. Substitute Θ = π/2 into the expression tan(Θ) - sec(Θ) to check if it results in an indeterminate form like 0/0 or ∞/∞.
2. Recognize that as Θ approaches π/2, tan(Θ) approaches ∞ and sec(Θ) also approaches ∞, leading to an indeterminate form of ∞ - ∞. To apply l'Hôpital's Rule, we need to rewrite the expression in a form suitable for the rule, typically as a fraction.
3. Rewrite the expression tan(Θ) - sec(Θ) as a single fraction. Recall that tan(Θ) = sin(Θ)/cos(Θ) and sec(Θ) = 1/cos(Θ). Combine these into a single fraction: (sin(Θ) - 1) / cos(Θ).
4. Now, check the new expression (sin(Θ) - 1) / cos(Θ) as Θ approaches π/2. Substitute Θ = π/2 to see if it results in an indeterminate form 0/0, which is suitable for l'Hôpital's Rule.
5. Apply l'Hôpital's Rule by differentiating the numerator and the denominator separately. Differentiate sin(Θ) - 1 to get cos(Θ), and differentiate cos(Θ) to get -sin(Θ). The new limit to evaluate is lim_Θ→π/2 (cos(Θ) / -sin(Θ)).

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. They are essential for understanding continuity, derivatives, and integrals. In this context, evaluating the limit as Θ approaches π/2 involves analyzing the behavior of the functions tan(Θ) and sec(Θ) near that point.

l'Hôpital's Rule

l'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if these forms occur, the limit of the ratio of two functions can be found by taking the derivative of the numerator and the derivative of the denominator. This rule simplifies the process of finding limits, especially when direct substitution leads to complications.

Trigonometric Functions

Trigonometric functions, such as tangent (tan) and secant (sec), are periodic functions that relate angles to ratios of sides in right triangles. Understanding their behavior, especially near critical points like π/2, is crucial for limit evaluation. In this case, tan(Θ) approaches infinity and sec(Θ) also approaches infinity as Θ approaches π/2, making the application of l'Hôpital's Rule relevant.