Determine the following limits.​


lim u→0^+ u − 1 / sin u

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding the function's behavior near points of interest, including points of discontinuity or infinity. In this case, we are interested in the limit as u approaches 0 from the positive side, which is crucial for evaluating the expression given.

Sine Function

The sine function, denoted as sin(u), is a periodic function that oscillates between -1 and 1. It is essential in calculus for analyzing limits, especially when dealing with small angles. As u approaches 0, sin(u) can be approximated by its Taylor series expansion, which is useful for simplifying expressions involving limits.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(u)/g(u) results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in the given limit problem, where direct substitution leads to an indeterminate form.