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/4 cot x−1 / x−π/4

Verified step by step guidance

1. Recognize that the given limit can be interpreted as the derivative of a function at a point. Specifically, it resembles the definition of the derivative of the function f(x) = cot(x) at the point x = π/4.
2. Recall the definition of the derivative: f'(a) = lim x→a (f(x) - f(a)) / (x - a). In this case, f(x) = cot(x) and a = π/4, so f(a) = cot(π/4) = 1.
3. Substitute f(x) = cot(x) and f(a) = 1 into the derivative definition: f'(π/4) = lim x→π/4 (cot(x) - 1) / (x - π/4). This matches the given limit expression.
4. To find the derivative f'(x) of f(x) = cot(x), use the derivative formula: f'(x) = -csc^2(x). Evaluate this derivative at x = π/4.
5. Calculate f'(π/4) using the derivative formula: f'(π/4) = -csc^2(π/4). Since csc(π/4) = √2, f'(π/4) = -(√2)^2 = -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 context, evaluating the limit as x approaches π/4 helps determine the behavior of the function cot(x) - 1 near that point.

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 problem, the limit can be interpreted as the derivative of the function cot(x) at x = π/4.

Cotangent Function

The cotangent function, denoted as cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). It is important to understand its behavior, especially around specific angles like π/4, where cot(π/4) equals 1. This knowledge is crucial for evaluating the limit in the given problem.