= Limit =

A '''limit''' is the value that a function approaches as the input approaches some value.
 
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== 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:

{{attachment:lim.svg}}



=== 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'':

{{attachment:limr.svg}}

A '''left-handed limit''' approaches ''a'' from the left and considers only the input values ''x < a'':

{{attachment:liml.svg}}

This is primarily useful for thinking of '''infinite limits'''. The limit of ''f(x)'' as ''x'' approaches positive infinity is notated as:

{{attachment:liminf1.svg}}

The limit of ''f(x)'' as ''x'' approaches negative infinity is notated as:

{{attachment:liminf2.svg}}



=== 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:

{{attachment:vec.svg}}

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 [[Calculus/Distance#Euclidean_distance|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:

{{attachment:mult1.svg}}

The limit is defined wherever ''ε'' can be made smaller by making ''δ'' smaller. It is notated as:

{{attachment:mult2.svg}}



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== Rewriting Limits ==

Limits can descend into continuous functions. That is, consider an inconvenient compound function like {{attachment:desc1.svg}}. Because exponentiation is a continuous function, this limit is equivalent to {{attachment:desc2.svg}}.



=== 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'''.

{{attachment:rule.svg}}

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 [[Calculus/Derivative|differentiable]].



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CategoryRicottone