Limit algebra

What follows is a list of algebra expressions that can be used for calculating limits. It is an extension of the usual algebra. We also mention indeterminate expressions here. They are dealt with in more detail here. Here you find a brief list of the limit algebra if you want just a summary.

All rules written here as limit algebra are supported by theorems proved in textbooks, for instance the rules ∞ + L = ∞ and 1/0⁺ = ∞ can be expressed as follows:

If f goes to infinity at a and g converges to a real number L at a, then f + g goes to infinity at a.
If f converges to 0 at a and f > 0 on some reduced neighborhood of a, then 1/f goes to infinity at a.

And now the rules.

Addition/subtraction:
Real numbers add/subtract in the usual way.
Infinity behaves nicely in most cases:
∞ ± L = ∞ for all real L,
∞ + ∞ = ∞.
Indeterminate expression: ∞ − ∞.

Multiplication/division:
Real numbers multiply in the usual way, also division works as usual as long as the denominator is not equal to zero.
1/0⁺ = ∞ and 1/0^- = −∞.
Infinity behaves nicely in the following cases:
∞⋅L = ∞ and ∞/L = ∞ for all positive L,
∞⋅L = −∞ and ∞/L = −∞ for all negative L,
L/∞ = 0 for all real L,
∞⋅∞ = ∞.
Indeterminate expressions: ∞⋅0, , and .
The expressions L/0 for a non-zero L and ∞/0 are usually not included among indeterminate expressions, but they do belong there, since they can yield three distinct answers: ∞, −∞ and DNE. It is enough to know the outcome of 1/0 (then for instance L/0=L⋅1/0) and there we have the two equalities above with 0⁺ and 0^-. If the zero is not of these two types, that is, if it (during the limiting procedure) keeps changing sign, then the limit 1/0 does not exist.

Powers:
Real numbers work in powers A^B in the usual way for positive A. When A is not positive, one has to be cautious, in particular 0⁰ is an indeterminate expression (see below).
Infinity behaves nicely in the following cases:
∞^L = ∞ for all positive L, ∞^L = 0 for all negative L,
L^∞ = ∞ if L > 1, L^∞ = 0 if |L| < 1, and L^∞ DNE if L < −1,
L^−∞ = 1/L^∞ = (1/L)^∞, so L^−∞ = 0 if |L| > 1, L^−∞ = ∞ if 0 < L < 1, and L^−∞ DNE if −1 < L < 0.
∞^∞ = ∞.
Indeterminate expressions: 1^∞, 0⁰, ∞⁰.

Important note: General powers have to be handled in the basic e^ln form (with the exception of simple cases that can be handled by the limit algebra above). For instance,

(0⁺)⁰ = e^0⋅ln(0+) = e^0⋅(−∞) = e^-0⋅∞,

To be able to evaluate limits one also needs to know the values of elementary functions at endpoints of intervals of their domains.

Dictionary:
e^∞ = ∞, therefore e^−∞ = 1/e^∞ = 1/∞ = 0.
ln(∞) = ∞ and ln(0⁺) = −∞.
sin(∞) DNE, cos(∞) DNE.
tan((π/2 + kπ)⁺) = −∞, tan((π/2 + kπ)^-) = ∞, cot((kπ)⁺) = ∞, cot((kπ)^-) = −∞,
arctan(∞) = π/2, arctan(−∞) = −π/2, arccot(∞) = 0, arccot(−∞) = −π.
sinh(∞) = ∞, sinh(−∞) = −∞, cosh(∞) = cosh(−∞) = ∞.
argtanh(∞) = 1, argtanh(−∞) = −1, argcoth(∞) = 1, argcoth(−∞) = −1, argcoth(0⁺) = ∞, argcoth(0^-) = −∞.

Note: The basic difference between a "real" algebra and the limit algebra is that here a number does not represent a fixed quantity, but the outcome of some process. We can thus imagine that, say, 3 is actually "almost 3". This explains why some things do not work the way one would expect. For instance, from the usual algebra we know that 1 to anything is 1. However, the expression 1^∞ represents an "almost 1" raised to a "really really huge number", and numbers that are not exactly one when raised to huge numbers can give anything from zero to infinity. The exact answer depends on the balance, on the relationship between those two "almost" in "almost 1" and "almost infinity".