Infinity
Infinity or 
is a symbol that represents a positive infinite quantity.
Details
can be entered as \[Infinity] or 
inf
. - In StandardForm, Infinity is printed as
. - Infinity is converted to DirectedInfinity[1].
- Certain arithmetic operations work with Infinity.
- NumberQ[Infinity] yields False.
Examples
open all close allBasic Examples (2)
Scope (7)
to enter 
Limit[1 / x, x -> ∞]
Series[1 / (x ^ 2 + x + 1), {x, Infinity, 5}]Infinite limits of integration and summation:
Integrate[Exp[-x ^ 2], {x, -Infinity, Infinity}]NIntegrate[Exp[-x ^ 4 / (x ^ 2 + 1)], {x, -Infinity, Infinity}]Sum[Exp[-n ^ 2], {n, -Infinity, Infinity}]NSum[Exp[-n ^ 4 / (n ^ 2 + 1)], {n, -Infinity, Infinity}]Extended arithmetic with infinite quantities:
Infinity Infinity2 Infinity - Infinity
In this case the result depends on the growth rates of the two terms:
Limit[2t - (t ^ 2 + 1), t -> Infinity]Limit[2t - (t + 1), t -> Infinity]Use infinite quantities in predicates:
10 ^ 100 < InfinityFind values of mathematical functions at infinities:
Exp[-Infinity]Exp[Infinity]For some functions there is no definite limit, but a limiting interval:
Sin[Infinity]Tan[Infinity]In this case no limiting interval exists:
Exp[I Infinity]
Mathematical functions can also return infinite quantities at finite points:
Log[0]Gamma[0]Cases finds matches only at the first level by default:
e = x Gamma[x] + (x + 1)Gamma[x ^ 2 + 1]Cases[e, _Gamma]Set the level to Infinity to include all levels:
Cases[e, _Gamma, Infinity]Applications (2)
Stirling's asymptotic approximation:
Series[n!, {n, Infinity, 2}]Setting the precision to Infinity will produce a rational approximation:
SetPrecision[3.14, Infinity]N[%]Some functions require infinite precision or exact input:
CylindricalDecomposition[y ^ 2 + 2.5 x ^ 2 < 1.5, {x, y}]
Apply SetPrecision to the whole expression to rationalize all inexact numbers:
CylindricalDecomposition[SetPrecision[y ^ 2 + 2.5 x ^ 2 < 1.5, Infinity], {x, y}]Neat Examples (1)
Use an infinite WorkingPrecision in Plot:
Reap[Plot[2x + 1, {x, 0, 1}, WorkingPrecision -> Infinity, EvaluationMonitor :> Sow[x], PlotPoints -> 3, MaxRecursion -> 0 ]]Plot[Numerator[x], {x, 0, 10}, WorkingPrecision -> Infinity, PlotPoints -> 1000, MaxRecursion -> 0 ]Related Links
MathWorld
The Wolfram Functions SiteText
Wolfram Research (1988), Infinity, Wolfram Language function, https://reference.wolfram.com/language/ref/Infinity.html (updated 1996).
CMS
Wolfram Language. 1988. "Infinity." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 1996. https://reference.wolfram.com/language/ref/Infinity.html.
APA
Wolfram Language. (1988). Infinity. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/Infinity.html