One-sided limit

Definition If I represents some interval that is contained in the domain of a function f and if a is a point in I, then the right-sided limit as x approaches a can be rigorously defined as the value R that satisfies: and the left-sided limit as x approaches a can be rigorously defined as the value L that satisfies: These definitions can be represented more symbolically as follows: Let I represent an interval, where I \subseteq \mathrm{domain}(f) and a \in I , then \begin{align} \lim_{x \to a^{+}} f(x) = R &\iff \forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I, 0 Intuition In comparison to the formal definition for the limit of a function at a point, the one-sided limit (as the name would suggest) only deals with input values to one side of the approached input value. For reference, the formal definition for the limit of a function at a point is as follows: : \lim_{x \to a} f(x) = L ~~~ \iff ~~~ \forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I, 0 To define a one-sided limit, we must modify this inequality. Note that the absolute distance between x and a is |x - a| = |(-1)(-x + a)| = |(-1)(a - x)| = |(-1)||a - x| = |a - x|. For the limit from the right, we want x to be to the right of a, which means that a , so x - a is positive. From above, x - a is the distance between x and a. We want to bound this distance by our value of \delta, giving the inequality x - a . Putting together the inequalities 0 and x - a and using the transitivity property of inequalities, we have the compound inequality 0 . Similarly, for the limit from the left, we want x to be to the left of a, which means that x . In this case, it is a - x that is positive and represents the distance between x and a. Again, we want to bound this distance by our value of \delta, leading to the compound inequality 0 . Now, when our value of x is in its desired interval, we expect that the value of f(x) is also within its desired interval. The distance between f(x) and L, the limiting value of the left sided limit, is |f(x) - L|. Similarly, the distance between f(x) and R, the limiting value of the right sided limit, is |f(x) - R|. In both cases, we want to bound this distance by \varepsilon, so we get the following: |f(x) - L| for the left sided limit, and |f(x) - R| for the right sided limit. ==Examples==

Example 1. The limits from the left and from the right of g(x) := - \frac{1}{x} as x approaches a := 0 are, respectively \lim_{x \to 0^-} -\frac{1}{x} = + \infty \qquad \text{ and } \qquad \lim_{x \to 0^+} {-1/x} = - \infty. The reason why \lim_{x \to 0^-} -\frac{1}{x} = + \infty is because x is always negative (since x \to 0^- means that x \to 0 with all values of x satisfying x ), which implies that - 1/x is always positive so that \lim_{x \to 0^-} -\frac{1}{x} diverges to + \infty (and not to - \infty) as x approaches 0 from the left. Similarly, \lim_{x \to 0^+} -\frac{1}{x} = - \infty since all values of x satisfy x > 0 (said differently, x is always positive) as x approaches 0 from the right, which implies that - 1/x is always negative so that \lim_{x \to 0^+} -\frac{1}{x} diverges to - \infty. Example 2. One example of a function with different one-sided limits is f(x) = \frac{1}{1 + 2^{-1/x}}, where the limit from the left is \lim_{x \to 0^-} f(x) = 0 and the limit from the right is \lim_{x \to 0^+} f(x) = 1. To calculate these limits, first show that \lim_{x \to 0^-} 2^{-1/x} = \infty \qquad \text{ and } \qquad \lim_{x \to 0^+} 2^{-1/x} = 0, which is true because \lim_{x \to 0^-} {-1/x} = + \infty and \lim_{x \to 0^+} {-1/x} = - \infty so that consequently, \lim_{x \to 0^+} \frac{1}{1 + 2^{-1/x}} = \frac{1}{1 + \displaystyle\lim_{x \to 0^+} 2^{-1/x}} = \frac{1}{1 + 0} = 1 whereas \lim_{x \to 0^-} \frac{1}{1 + 2^{-1/x}} = 0 because the denominator diverges to infinity; that is, because \lim_{x \to 0^-} 1 + 2^{-1/x} = \infty. Since \lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x), the limit \lim_{x \to 0} f(x) does not exist. ==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. Alternatively, one may consider the domain with a half-open interval topology. ==Abel's theorem==

A noteworthy theorem treating one-sided limits of certain power series at the boundaries of their intervals of convergence is Abel's theorem. == Notes ==