One-sided limit
Encyclopedia
In calculus
, a one-sided limit is either of the two limits
of a function
f(x) of a real
variable x as x approaches a specified point either from below or from above. One should write either:
or 
or 
for the limit as x decreases in value approaching a (x approaches a "from the right" or "from above"), and similarly
or 
or 
for the limit as x increases in value approaching a (x approaches a "from the left" or "from below").
The two one-sided limits exist and are equal if and only if the limit of f(x) as x approaches a exists. In some cases in which the limit
does not exist, the two one-sided limits nonetheless exist. Consequently the limit as x approaches a is sometimes called a "two-sided limit". In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.
The left-sided limit can be rigorously defined as:
Similarly, the right-sided limit can be rigorously defined as:
Where
represents some interval that is within the domain of 
whereas
is Abel's theorem
.
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
, a one-sided limit is either of the two limits
Limit of a function
In mathematics, the limit of a function is a fundamental concept in calculus and analysis concerning the behavior of that function near a particular input....
of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f(x) of a real
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
variable x as x approaches a specified point either from below or from above. One should write either:
or or for the limit as x decreases in value approaching a (x approaches a "from the right" or "from above"), and similarly
or or for the limit as x increases in value approaching a (x approaches a "from the left" or "from below").
The two one-sided limits exist and are equal if and only if the limit of f(x) as x approaches a exists. In some cases in which the limit
does not exist, the two one-sided limits nonetheless exist. Consequently the limit as x approaches a is sometimes called a "two-sided limit". In some cases one of the two one-sided limits exists and the other does not, and in some cases neither exists.
The left-sided limit can be rigorously defined as:
Similarly, the right-sided limit can be rigorously defined as:
Where
represents some interval that is within the domain of Examples
One example of a function with different one-sided limits is the following:whereas
Relation to topological definition of limit
The one-sided limit to a point p corresponds to the general definition of limit, with the domain of the function restricted to one side, by either allowing that the function domain is a subset of the topological space, or by considering a one-sided subspace, including p.Abel's theorem
A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergenceRadius of convergence
In mathematics, the radius of convergence of a power series is a quantity, either a non-negative real number or ∞, that represents a domain in which the series will converge. Within the radius of convergence, a power series converges absolutely and uniformly on compacta as well...
is Abel's theorem
Abel's theorem
In mathematics, Abel's theorem for power series relates a limit of a power series to the sum of its coefficients. It is named after Norwegian mathematician Niels Henrik Abel.-Theorem:...
.