A First Course in Complex Analysis · Problem 6

Examples of Functions — Beck, Marchesi, Pixton & Sabalka

Mathematics — Complex Analysis · complex-limits

Problem

Proposition 2.1.4 is useful for showing that limits do not exist, but it is not at all useful for showing that a limit does exist. For example, define f(z)=x2yx4+y2wherez=x+iy≠0.f(z) = \frac{x^2 y}{x^4 + y^2} \quad \text{where} \quad z = x + iy \neq 0. Show that the limits of ff at 00 along all straight lines through the origin exist and are equal, but lim⁡z→0f(z)\lim_{z \to 0} f(z) does not exist. Hint. Consider the limit along the parabola y=x2y = x^2.

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