Explore the complexities of limits in pure mathematics through this 25-minute lecture that challenges common misconceptions. Delve into the three distinct types of limiting behavior in infinite series, with a focus on the Cauchy condition's significance. Examine the process of converting series to sequences, understand the limits of series and sequences, and critically analyze the convergence of Cauchy sequences. Uncover the controversial practice of creating limits without proper justification, gaining valuable insights into the often-overlooked intricacies of mathematical analysis.