Problem 1
It compares late terms to one another without assuming the limit is already known.
Faculty solution guide
Cauchy–Riemann Equations
Accept equivalent mathematically correct reasoning. Require explicit definitions, theorem hypotheses, proof structure, verification, and limitations.
Problem 2
Identify the first invalid inference, missing hypothesis, quantifier error, unjustified limit interchange, or unproved existence claim; then supply a valid repair.
Problem 3
A complete response explicitly states the definitions, domain, hypotheses, exact target, and theorem dependencies for Cauchy–Riemann Equations.
Problem 4
The argument must address deriving necessary conditions and applying sufficient regularity conditions with each nontrivial step justified and the final conclusion restated.
Problem 5
A strong application explains the mathematical model in fluid flow, electromagnetism, signal processing, geometry, and dynamical systems, verifies conditions, and names at least one limitation.
Problem 6
Accept a second valid proof, a counterexample showing necessity of a hypothesis, an independent estimate, or a reproducible finite check that is correctly described as supporting evidence.