Did Weierstrass’s differential calculus have a limit-avoiding character? His definition of a limit in ϵ – δ style

In the 1820s, Cauchy founded his calculus on his original limit concept and developed his theory by using inequalities, but he did not apply these inequalities consistently to all parts of his theory. In contrast, Weierstrass consistently developed his 1861 lectures on differential calculus in terms of epsilonics. His lectures were not based on Cauchy’s limit and are distinguished by their limit-avoiding character. Dugac’s partial publication of the 1861 lectures makes these differences clear. But in the unpublished portions of the lectures, Weierstrass actually defined his limit in terms of inequalities. Weierstrass’s limit was a prototype of the modern limit but did not serve as a foundation of his calculus theory. For this reason, he did not provide the basic structure for the modern style analysis. Thus it was Dini’s 1878 textbook that introduced the definition of a limit in terms of inequalities.