Limits at Infinity

Complex Analysis · Axiom Academy

In the complex plane there is just ONE point at infinity — approached from every direction at once. 1. One Infinity, Every Direction Send z off to infinity — but in there is no single road out. z can leave along the positive reals, straight up the imaginary axis, on a diagonal, on a spiral. We write to mean only one thing: |z| grows without bound, no matter the direction . means the magnitude blows up — the angle is free the limit L is the single value every direction lands on means: for every there is an R > 0 with . One bound R controls all directions at once. Here is the one move that tames infinity. The map w = 1/z turns "far out" into "near 0 ": if |z| is huge then |w| = 1/|z| is tiny. So becomes the ordinary limit , and the single point at infinity is just the origin in the w -plane. study the behavior at by studying f(1/w) at 0 Take . Substituting gives f(1/w) = w , and as . So : the function z has a pole at , and 1/z has a zero there. 3. Rational Functions, Read by Degree For a quotient of polynomials the answer is decided entirely by the two leading terms. Push z outward and every lower-order term becomes negligible; only the highest powers on top and bottom survive. Three cases — that is the whole story. : the denominator wins — the limit is 0 . : a fair race — the limit is the ratio of leading coefficients. : the numerator wins — the limit is . Worked example — the equal-degree case Evaluate . Substitute w = 1/z :

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