Limit to Infinity of Error Function
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Theorem
- $\ds \lim_{x \mathop \to \infty} \map \erf x = 1$
where $\erf$ denotes the error function.
Proof
| \(\ds \lim_{x \mathop \to \infty} \map \erf x\) | \(=\) | \(\ds \lim_{x \mathop \to \infty} \paren {\frac 2 {\sqrt \pi} \int_0^x e^{-t^2} \rd t}\) | Definition of Error Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \int_0^\infty e^{-t^2} \rd t\) | Multiple Rule for Limits of Real Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 2 {\sqrt \pi} \times \frac {\sqrt \pi} 2\) | Integral to Infinity of Exponential of -t^2 | |||||||||||
| \(\ds \) | \(=\) | \(\ds 1\) |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 35$: Error Function $\ds \map \erf x = \frac 2 {\sqrt \pi} \int_0^x e^{-u^2} \rd u$: $35.3$