How does one go about evaluating the limits of nonelementary antiderivatives of functions? We obvio

crossoverman9b

crossoverman9b

Answered question

2022-06-25

How does one go about evaluating the limits of nonelementary antiderivatives of functions?
We obviously know the following to be equal:
lim z ± erf ( z ) = ± 1
But as far as I'm aware, the only way we know how to calculate any given value of the error function is to approximate it. How do we know the limits at infinities without any way to evaluate it algabraically?

Answer & Explanation

marktje28

marktje28

Beginner2022-06-26Added 22 answers

Step 1
lim x ± erf ( x ) = lim x ± 2 π 0 x e t 2 d t = 2 π 0 ± e t 2
using the fact e t 2 is an even function:
± 2 π 0 e t 2
Step 2
The integral is known as the gaussian integral(half of it) and is known to equal π 2 and thus,
lim z ± erf ( z ) = ± 1

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