e^(-1/x^2)

Input

e^(-1/x^2)

Plot

Plot
Plot

Roots

(no roots exist)

Properties as a real function

Domain

{x element R : x!=0}

Range

{y element R : 0<y<1}

Parity

even

Series expansion at x=∞

1 - (1/x)^2 + 1/(2 x^4) + O((1/x)^6) (Laurent series)
Big‐O notation »

Derivative

d/dx(e^(-1/x^2)) = (2 e^(-1/x^2))/x^3

Indefinite integral

integral e^(-1/x^2) dx = sqrt(π) erf(1/x) + e^(-1/x^2) x + constant

Limit

lim_(x-> ± ∞) e^(-1/x^2) = 1

Alternative representation

e^(-1/x^2) = w^a for a = -1/(x^2 log(w))
e^(-1/x^2) = z^(-1/x^2) for z = e
e^(-1/x^2) = 1 + 2/(-1 + coth(-1/(2 x^2)))
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