Properties of Complex Limits

Complex Analysis · Axiom Academy

LESSON Properties of Complex Limits Algebraic rules for computing limits — the same laws that worked for real functions carry over, so you can find most limits by substitution. Suppose two functions settle on their own targets near z_0 : and . The limit laws say that any algebraic combination of f and g settles on the same combination of A and B . Watch the two limits land, then watch their sum and product land exactly where the laws predict. . The limit of a sum or difference is the sum or difference of the limits. . The limit of a product is the product of the limits. for any constant . Constants pull straight out of the limit. provided , and for any positive integer n . 2. Polynomial Limits by Substitution Start from two facts the laws hand you for free: (a constant) and (the identity). Build any polynomial out of these with the sum, product, and power rules, and the limit is just the polynomial evaluated at z_0 . In the animation, as z spirals into z_0 = i , the value P(z)=z^2+2z-1 is dragged straight to P(i) . Using the limit laws, we simply substitute z = i : 3. Rational Limits & the Modulus Rule The quotient rule lets us substitute into a rational function P(z)/Q(z) — as long as the denominator does not vanish . The animation tracks both pieces of as : the numerator slides toward 0 , the denominator toward -2 (safely nonzero), so the ratio lands on 0 . If then , since the modulus is continuous. Since the denominator approaches : Caution: the modulus converse is false

This is the written version of the interactive lesson above. See the full Complex Analysis course.