Limit¶

Computes the limit of an expression as a variable approaches a value.

$ wo 'Limit[(1 + 1/n)^n, n -> Infinity]'
E

$ wo 'Limit[(Sin[x] - x)/x^3, x -> 0]'
-1/6

The limit at infinity of a quotient of power sums is decided from the leading exponents, so terms with fractional powers are handled exactly rather than by sampling:

$ wo 'Limit[x/(x + Sqrt[x]), x -> Infinity]'
1

$ wo 'Limit[(3 x^(3/2) + x)/(2 x^(3/2) - 1), x -> Infinity]'
3/2

Symbolic coefficients carry through:

$ wo 'Limit[(a x + b)/(c x + d), x -> Infinity]'
a/c

At the origin it is the smallest exponent that dominates, so quotients with fractional powers resolve there too:

$ wo 'Limit[(x + Sqrt[x])/(x - Sqrt[x]), x -> 0]'
-1

A two-sided limit only diverges when both sides agree, which for 1/x^k means an even integer k:

$ wo 'Limit[1/x^2, x -> 0]'
Infinity

$ wo 'Limit[1/x^3, x -> 0]'
Indeterminate

Naming a direction gives the one-sided answer:

$ wo 'Limit[1/x, x -> 0, Direction -> "FromAbove"]'
Infinity