Evaluate the following limits or state that they do not exist. (Hint: Identify each limit as the derivative of a function at a point.)
lim x→π/2 cos x/x−π/2

Verified step by step guidance

1. Recognize that the given limit can be interpreted as the derivative of a function at a specific point. The expression lim x→π/2 (cos x)/(x−π/2) resembles the definition of a derivative.
2. Recall the definition of the derivative of a function f at a point a: f'(a) = lim x→a (f(x) - f(a))/(x - a).
3. Identify the function f(x) = cos x and the point a = π/2. Notice that f(a) = cos(π/2) = 0.
4. Rewrite the limit in the form of the derivative definition: lim x→π/2 (cos x - cos(π/2))/(x - π/2). This matches the derivative definition for f(x) = cos x at x = π/2.
5. Conclude that the limit represents the derivative of f(x) = cos x at x = π/2, which is f'(π/2). To find this derivative, compute f'(x) = -sin x and evaluate it at x = π/2.

Key Concepts

Limits

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

Derivatives

The derivative of a function at a point measures the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function over an interval as the interval approaches zero. In this context, recognizing the limit as a derivative allows for the application of derivative rules to evaluate the limit.

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 the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator separately. This rule is particularly useful in simplifying complex limit problems.