In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a limiting case of a
mathematical object is a
special case that arises when one or more components of the object take on their most extreme possible values.
For example:
* In
statistics
Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a s ...
, the limiting case of the
binomial distribution
In probability theory and statistics, the binomial distribution with parameters and is the discrete probability distribution of the number of successes in a sequence of statistical independence, independent experiment (probability theory) ...
is the
Poisson distribution. As the number of events tends to infinity in the binomial distribution, the random variable changes from the binomial to the Poisson distribution.
*A
circle is a limiting case of various other figures, including the
Cartesian oval, the
ellipse, the
superellipse, and the
Cassini oval. Each type of figure is a circle for certain values of the defining parameters, and the generic figure appears more like a circle as the limiting values are approached.
*
Archimedes
Archimedes of Syracuse ( ; ) was an Ancient Greece, Ancient Greek Greek mathematics, mathematician, physicist, engineer, astronomer, and Invention, inventor from the ancient city of Syracuse, Sicily, Syracuse in History of Greek and Hellenis ...
calculated an approximate value of
π by treating the circle as the limiting case of a
regular polygon with 3 × 2
''n'' sides, as ''n'' gets large.
*In
electricity and magnetism, the
long wavelength limit is the limiting case when the
wavelength is much larger than the system size.
*In
economics
Economics () is a behavioral science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services.
Economics focuses on the behaviour and interac ...
, two limiting cases of a
demand curve or
supply curve are those in which the
elasticity is zero (the totally inelastic case) or infinity (the infinitely elastic case).
*In
finance
Finance refers to monetary resources and to the study and Academic discipline, discipline of money, currency, assets and Liability (financial accounting), liabilities. As a subject of study, is a field of Business administration, Business Admin ...
,
continuous compounding is the limiting case of compound interest in which the compounding period becomes infinitesimally small, achieved by taking the limit as the number of compounding periods per year goes to infinity.
A limiting case is sometimes a
degenerate case in which some qualitative properties differ from the corresponding
properties of the generic case. For example:
*A
point is a degenerate
circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is cal ...
, whose
radius
In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...
is zero.
*A
parabola
In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...
can degenerate into two distinct or coinciding
parallel lines.
*An
ellipse can degenerate into a single point or a
line segment
In geometry, a line segment is a part of a line (mathematics), straight line that is bounded by two distinct endpoints (its extreme points), and contains every Point (geometry), point on the line that is between its endpoints. It is a special c ...
.
*A
hyperbola can degenerate into two
intersecting lines.
See also
*
Degeneracy (mathematics)
*
Limit (mathematics)
In mathematics, a limit is the value that a function (or sequence) approaches as the argument (or index) approaches some value. Limits of functions are essential to calculus and mathematical analysis, and are used to define continuity, deriv ...
References
Mathematical concepts
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