Limit to Infinity of Complementary Error Function

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Theorem

$\ds \lim_{x \mathop \to \infty} \map \erfc x = 0$

where $\erfc$ denotes the complementary error function.


Proof

\(\ds \lim_{x \mathop \to \infty} \map \erfc x\) \(=\) \(\ds \lim_{x \mathop \to \infty} \paren {1 - \map \erf x}\) Definition of Complementary Error Function
\(\ds \) \(=\) \(\ds 1 - \lim_{x \mathop \to \infty} \map \erf x\) Sum Rule for Limits of Real Functions
\(\ds \) \(=\) \(\ds 1 - 1\) Limit to Infinity of Error Function
\(\ds \) \(=\) \(\ds 0\)

$\blacksquare$


Sources