Limit
A limit is the value that a function approaches as the input approaches some value.
Contents
Description
Consider the limit of f(x) as the input value approaches a. There is a range of input values around a defined by being no more than δ away; this can almost be described as [a-δ,a+δ] but it excludes a itself. This range of input values maps to a constrained set of output values that are at most ε away from the limit of f(x) as it approaches a; |f(x) - L| < ε. A limit is defined wherever ε can be made smaller by making δ smaller.
That is to say, if there are discontinuities preventing ε from being made any smaller, the limit is not defined.
A limit is notated as:
One-sided
In certain circumstances, it is useful to consider one-sided limits. This can be a solution to describing discontinuous function, in that input values to the other side do not need to converge on the limit.
A right-handed limit approaches a from the right, and considers only the input values a < x:
A left-handed limit approaches a from the left and considers only the input values x < a:
This is primarily useful for thinking of infinite limits. The limit of f(x) as x approaches positive infinity is notated as:
The limit of f(x) as x approaches negative infinity is notated as:
Vector-Valued Function
The limit of the vector-valued function r(t) = f(t)i + g(t)j + h(t)k as t approaches a is given by:
provided that the component limits exist.
Multivariate Function
Instead of a range of input values mapping to a constrained set of outputs, consider a disk (or ball). The disk (or ball) is formed by all input coordinates defined by being having a distance from a less than δ, but also not including a itself. The definition of a limit, and the description of when a limit exists, then follows the univariate case as above.
Consider the limit of f(x,y) as it approaches (a,b). There is a disk of (x,y) coordinates around (a,b) such that:
The limit is defined wherever ε can be made smaller by making δ smaller. It is notated as:
Rewriting Limits
Limits can descend into continuous functions. That is, consider an inconvenient compound function like 
. Because exponentiation is a continuous function, this limit is equivalent to 
.
L'Hôpital's Rule
If the function being evaluated can be expressed as a ratio of two function (i.e., f/g), it may be possible to calculate the limit using L'Hôpital's rule.
If the limit ordinarily evaluates to 0/0, or ∞/∞, or -∞/-∞, then this rule is applicable. Importantly the signs must agree; ∞/-∞ is not applicable.
In addition, the two functions must be differentiable.