Determine the following limits.​


lim x→π/2 1/√sin x − 1 / x + π/2

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 question, evaluating the limit as x approaches π/2 involves analyzing the behavior of the function near that point.

Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit's value. In this case, substituting x = π/2 into the limit expression results in an indeterminate form, necessitating further analysis, such as algebraic manipulation or L'Hôpital's Rule.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms. It states that if the limit of f(x)/g(x) results in 0/0 or ∞/∞, the limit 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 in complex expressions like the one presented in the question.