Pointwise limit ($f(x)={\lim }_{n\to \infty }{f}_n(x)$)

A pointwise limit assigns to each input the limit of a sequence of function values there, on the specified domain of a function. It is unique wherever it exists, by uniqueness of the numerical limit of a sequence. Its existence supplies no common approximation rate over the domain of a function. For $f_n(x)=x^n$ on $[0,1]$, it equals zero below one and one at one; restricting to $[0,a]$ with $a<1$ gives the zero function.