List of basic formulas and concepts related to the limits of functions of a complex variable:

  1. Definition of limit:
    [
    \lim_{z \to z_0} f(z) = L
    ]
    If for any (\epsilon > 0) there exists (\delta > 0) such that if (0 < |z – z_0| < \delta), then (|f(z) – L| < \epsilon).
  2. One-sided limits:
  • Left-hand limit:
    [
    \lim_{z \to z_0^-} f(z) = L
    ]
  • Right-hand limit:
    [
    \lim_{z \to z_0^+} f(z) = L
    ]
  1. Limits at infinity:
    [
    \lim_{|z| \to \infty} f(z) = L
    ]
    If for any (\epsilon > 0) there exists (R > 0) such that if (|z| > R), then (|f(z) – L| < \epsilon).
  2. Properties of limits:
  • Linearity:
    [
    \lim_{z \to z_0} (af(z) + bg(z)) = a \lim_{z \to z_0} f(z) + b \lim_{z \to z_0} g(z)
    ]
    where (a) and (b) are constants.
  • Product:
    [
    \lim_{z \to z_0} (f(z)g(z)) = \lim_{z \to z_0} f(z) \cdot \lim_{z \to z_0} g(z)
    ]
    provided that both limits exist.
  • Private:
    [
    \lim_{z \to z_0} \frac{f(z)}{g(z)} = \frac{\lim_{z \to z_0} f(z)}{\lim_{z \to z_0 } g(z)}
    ]
    provided that (\lim_{z \to z_0} g(z) \neq 0).
  1. Limits of composite functions:
    If (g(z)) is continuous at (z_0) and (\lim_{z \to z_0} f(z) = L), then:
    [
    \lim_{z \to z_0} g(f(z)) = g(L)
    ]

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