Let f be a real-valued function such that f(x) is defined when x is near the number a. If values of f(x) can be constrained to be arbitrarily close to a number L by requiring x to be sufficiently close (but not equal) to a (on either side of a), then the (two-sided) limit of f(x) as x approaches a is L, denotedlim_(x->a) f(x) = L. A more precise formulation known as the epsilon-delta definition of a limit lets f be a real-valued function defined on an open interval containing a, except possibly at a itself. Then the limit of f(x) as x approaches a is L if for every number ϵ>0 there is a number δ>0 such that if 0

function | two-sided limit of a function

derivative | limit of a function from the left | limit of a function from the right | limit of a sequence | one-sided limit of a function

Bernard Placidus Johann Nepomuk Bolzano | Augustin-Louis Cauchy | Karl Weierstrass