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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Complementary Error Function $\ds \map \erfc x = 1 - \map \erf x = \frac 2 {\sqrt \pi} \int_x^\infty e^{-u^2} \rd u$: $35.6$