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Full Idea
After Weierstrass had stressed the importance of limits, one now needed to be able to prove the existence of such limits.
Gist of Idea
Weierstrass made limits central, but the existence of limits still needed to be proved
Source
report of Karl Weierstrass (works [1855]) by David Bostock - Philosophy of Mathematics 4.4
Book Ref
Bostock,David: 'Philosophy of Mathematics: An Introduction' [Wiley-Blackwell 2009], p.98
A Reaction
The solution to this is found in work on series (going back to Cauchy), and on Dedekind's cuts.