Antilimits are used to apply the methods of sequence transformations to divergent series. There is no unique definition, but typically if the sum $\sum_{n}a_n(x)$ converges to some function $f(x)$ for $|x|<\rho$, and this function can be continued analytically for $|x|>\rho$, then $f(x)$ is called the limit of the divergent series for $|x|<\rho$ and the antilimit for $|x|>\rho$.
A classic application (from Christopher Small's Expansions and Asymptotics for Statistics) is the proof of the identity
$$\frac{1}{1-x}+\frac{1}{1+x^2}-\frac{x}{1-x^2}=\frac{2}{1-x^4}$$
by sequence transformations:
$$\frac{1}{1-x}+\frac{1}{1+x^2}-\frac{x}{1-x^2}=(1+x+x^2+\cdots)+$$ $$(1-x^2+x^4-\cdots)-(x+x^3+x^5+\cdots)$$ $$=2+2x^4+2x^8+\cdots=\frac{2}{1-x^4}.$$
Although the sums of the series only have a limit for $|x|<1$, the proof of the identity remains valid for $|x|>1$ if the sums are interpreted as antilimits.