cot1x

Cot1x is not a standard universally accepted notation, but in many mathematical contexts it is used to denote the cotangent of the reciprocal of x, i.e., cot(1/x). Since cot(y) = cos(y)/sin(y), cot(1/x) = cos(1/x)/sin(1/x).

Domain and singularities: The function is defined for real x except where sin(1/x) = 0, because cot(y) is undefined when sin(y) = 0. This occurs when 1/x = nπ for some integer n, i.e., at x = 1/(nπ). Thus the domain is R \ {0} \ {1/(nπ) : n ∈ Z\{0}}. At each x = 1/(nπ) the function has a simple vertical asymptote (pole).

Behavior and limits: The cotangent function has a period of π in its argument, but cot(1/x) as a function of x is highly oscillatory near x = 0 due to the unbounded variation of 1/x. As x → ∞ or x → −∞, y = 1/x → 0, and cot(y) behaves like 1/y, so cot(1/x) grows roughly linearly with x for large |x|. As x approaches a pole x = 1/(nπ), cot(1/x) diverges to ±∞. The overall range on its domain is all real numbers, since cot(y) attains every real value for y ∈ (0, π) and the argument 1/x sweeps through such values away from the poles.

Notes: If read as cot(x) instead of cot(1/x), the function would be the standard cotangent with a different domain and set of discontinuities. Because notation can vary, cot1x should be interpreted explicitly as cot(1/x) unless stated otherwise.