|
Size: 1301
Comment: Ball
|
Size: 2611
Comment: Rewrite
|
| Deletions are marked like this. | Additions are marked like this. |
| Line 13: | Line 13: |
| The limit of ''f'' as ''x'' approaches infinity is notated as: | 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. |
| Line 15: | Line 15: |
| {{attachment:lim1.svg}} | 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}} |
| Line 31: | Line 57: |
| One way to express the limit of a multivariate function is to imagine a disk (or ball) around a target point. The disk (ball) is principally formed by all input coordinates no more than ''δ'' away from the target coordinate (leaving out the target itself). This set of input coordinates map to a set of outputs characterized by ''|f(·) - L| < ε'', i.e. these outputs are at most ''ε'' away from the limit. As ''δ'' shrinks to 0, so too does the set of input coordinates, and so too does ''ε''. | 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. |
| Line 37: | Line 63: |
| The limit is found by shrinking ''δ''. This is notated as: | The limit is defined wherever ''ε'' can be made smaller by making ''δ'' smaller. It is notated as: |
Limit
A limit is the value that a function approaches as the input approaches some value.
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-Values Function
The limit of the vector-values 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:
