Complex Analysis: Theory and Applications · Problem 2.5.3

Analytic and Harmonic Functions — Howell & Matthews

Mathematics — Complex Analysis · limits-at-infinity

Problem

The function f(z)f(z) is said to have the limit LL as zz approaches ∞\infty, and we write lim⁡z→∞f(z)=L\lim_{z\to\infty}f(z) = L iff for every ε>0\varepsilon > 0 there exists an R>0R > 0 such that f(z)∈Dε(L)f(z) \in D_\varepsilon(L) (i.e., ∣f(z)−L∣<ε|f(z) - L| < \varepsilon) whenever ∣z∣>R|z| > R. Likewise, lim⁡z→z0f(z)=∞\lim_{z\to z_0}f(z) = \infty iff for every R>0R > 0 there exists δ>0\delta > 0 such that ∣f(z)∣>R|f(z)| > R whenever z∈Dδ∗(z0)z \in D^*_\delta(z_0) (i.e., 0<∣z−z0∣<δ0 < |z - z_0| < \delta). Use this definition to show the following: (a) that lim⁡z→∞1z=0\displaystyle\lim_{z\to\infty}\frac{1}{z} = 0; (b) that lim⁡z→01z=∞\displaystyle\lim_{z\to 0}\frac{1}{z} = \infty.

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