Difference between revisions of "Limit of a function"
Karl Jones (Talk | contribs) (Created page with "In mathematics, the '''limit of a function''' is a fundamental concept in calculus and mathematical analysis concerning the behavior of that Function (mathematic...") |
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On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist. | On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist. | ||
| − | The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function. It also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function. | + | The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function. |
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| + | It also appears in the definition of the [[derivative]]: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function. | ||
== See also == | == See also == | ||
Latest revision as of 10:52, 6 September 2016
In mathematics, the limit of a function is a fundamental concept in calculus and mathematical analysis concerning the behavior of that function near a particular input.
Description
Formal definitions, first devised in the early 19th century, are given below. Informally, a function f assigns an output f(x) to every input x. We say the function has a limit L at an input p: this means f(x) gets closer and closer to L as x moves closer and closer to p.
More specifically, when f is applied to any input sufficiently close to p, the output value is forced arbitrarily close to L.
On the other hand, if some inputs very close to p are taken to outputs that stay a fixed distance apart, we say the limit does not exist.
The notion of a limit has many applications in modern calculus. In particular, the many definitions of continuity employ the limit: roughly, a function is continuous if all of its limits agree with the values of the function.
It also appears in the definition of the derivative: in the calculus of one variable, this is the limiting value of the slope of secant lines to the graph of a function.
See also
- Big O notation
- Dependent and independent variables
- Function (mathematics)
- l'Hôpital's rule
- Limit of a sequence
- List of limits
- Limit superior and limit inferior
- Net (topology)
- Non-standard calculus
- One-sided limit
- Squeeze theorem
- Special trig limits
External links
- Limit of a function @ Wikipedia]
