fxsin1x

fxsin1x is a shorthand notation used in mathematics to denote the product of a function f with the sine of the reciprocal, commonly written as f(x) sin(1/x). It is a simple yet instructive example in real analysis, often employed to study limits, continuity, and oscillatory behavior near x = 0. The expression can be defined on a domain that includes points near zero, with the value at x = 0 specified separately if needed.

The key feature of fxsin1x is the oscillation of sin(1/x) as x approaches zero. Because sin(1/x) oscillates between -1 and 1 without settling to a single value, the limit of f(x) sin(1/x) at 0 depends on how f(x) behaves near zero. If lim_{x->0} f(x) = 0, then lim_{x->0} f(x) sin(1/x) = 0 by the squeeze theorem. If lim_{x->0} f(x) = L ≠ 0 or if f(x) has no limit at 0, the limit of the product typically does not exist due to the ongoing oscillation of sin(1/x). For example, if f(x) = x, the product tends to 0; if f(x) ≡ 1, the limit does not exist.

In practice, fxsin1x is used to illustrate how multiplication by an oscillatory factor can affect limit and continuity properties, and it can be extended or modified by considering variants such as multiplying by cos(1/x) or by other slowly varying functions.