One-sided limit

thumb|350px|At <math>x = 0,</math> the function <math>f(x) = x^2 + \operatorname{sign}(x),</math> where <math>\operatorname{sign}(x)</math> denotes the [[sign function, has a left limit of <math>-1,</math> a right limit of <math>+1,</math> and a function value of <math>0.</math>]]

In calculus, a one-sided limit refers to either one of the two limits of a function <math>f(x)</math> of a real variable <math>x</math> as <math>x</math> approaches a specified point either from the left or from the right.

The limit, as <math>x</math> decreases in value approaching <math>a</math> (<math>x</math> approaches <math>a</math> "from the right" or "from above"), is denoted: or "from below"), is denoted:

and the left-sided limit as <math>x</math> approaches <math>a</math> can be rigorously defined as the value <math>L</math> that satisfies:

These definitions can be represented more symbolically as follows: Let <math>I</math> represent an interval, where <math>I \subseteq \mathrm{domain}(f)</math> and <math>a \in I </math>, then

<math display=block> \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 < x - a < \delta \longrightarrow | f(x) - R | < \varepsilon, \\

\lim_{x \to a^{- f(x) = L &\iff \forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I, 0 < a - x < \delta \longrightarrow | f(x) - L | < \varepsilon.

\end{align}

</math>

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:

:<math display=block>

\lim_{x \to a} f(x) = L

~~~ \iff ~~~

\forall \varepsilon \in \mathbb{R}_{+}, \exists \delta \in \mathbb{R}_{+}, \forall x \in I,

0 < |x - a| < \delta \implies | f(x) - L | < \varepsilon

.</math>

To define a one-sided limit, we must modify this inequality. Note that the absolute distance between <math>x</math> and <math>a</math> is

For the limit from the right, we want <math>x</math> to be to the right of <math>a</math>, which means that <math>a < x</math>, so <math>x - a</math> is positive. From above, <math>x - a</math> is the distance between <math>x</math> and <math>a</math>. We want to bound this distance by our value of <math>\delta</math>, giving the inequality <math>x - a < \delta</math>. Putting together the inequalities <math>0 < x - a</math> and <math>x - a < \delta</math> and using the transitivity property of inequalities, we have the compound inequality <math>0 < x - a < \delta </math>.

Similarly, for the limit from the left, we want <math>x</math> to be to the left of <math>a</math>, which means that <math>x < a</math>. In this case, it is <math>a - x</math> that is positive and represents the distance between <math>x</math> and <math>a</math>. Again, we want to bound this distance by our value of <math>\delta</math>, leading to the compound inequality <math>0 < a - x < \delta </math>.

Now, when our value of <math>x</math> is in its desired interval, we expect that the value of <math>f(x)</math> is also within its desired interval. The distance between <math>f(x)</math> and <math>L</math>, the limiting value of the left sided limit, is <math>|f(x) - L|</math>. Similarly, the distance between <math>f(x)</math> and <math>R</math>, the limiting value of the right sided limit, is <math>|f(x) - R|</math>. In both cases, we want to bound this distance by <math>\varepsilon</math>, so we get the following: <math>|f(x) - L| < \varepsilon</math> for the left sided limit, and <math>|f(x) - R| < \varepsilon</math> for the right sided limit.

Examples

Example 1. The limits from the left and from the right of <math display="inline"> g(x) := - \frac{1}{x}</math> as <math>x</math> approaches <math>a := 0</math> are, respectively

<math display=block> \lim_{x \to 0^-} -\frac{1}{x} = + \infty \qquad \text{ and } \qquad \lim_{x \to 0^+} {-1/x} = - \infty.</math>

The reason why <math display="inline">\lim_{x \to 0^-} -\frac{1}{x} = + \infty</math> is because <math>x</math> is always negative (since <math>x \to 0^-</math> means that <math>x \to 0</math> with all values of <math>x</math> satisfying <math>x < 0</math>), which implies that <math>- 1/x</math> is always positive so that <math display="inline">\lim_{x \to 0^-} -\frac{1}{x}</math> diverges to <math>+ \infty</math> (and not to <math>- \infty</math>) as <math>x</math> approaches <math>0</math> from the left.

Similarly, <math display="inline">\lim_{x \to 0^+} -\frac{1}{x} = - \infty</math> since all values of <math>x</math> satisfy <math>x > 0</math> (said differently, <math>x</math> is always positive) as <math>x</math> approaches <math>0</math> from the right, which implies that <math>- 1/x</math> is always negative so that <math display="inline">\lim_{x \to 0^+} -\frac{1}{x}</math> diverges to <math>- \infty.</math>

thumb|350px|Plot of the function <math display="inline"> f(x) = \frac{1}{1 + 2^{-1/x</math>.

Example 2. One example of a function with different one-sided limits is <math display="inline">f(x) = \frac{1}{1 + 2^{-1/x</math>, where the limit from the left is <math>\lim_{x \to 0^-} f(x) = 0</math> and the limit from the right is <math>\lim_{x \to 0^+} f(x) = 1.</math> To calculate these limits, first show that

<math display=block>\lim_{x \to 0^-} 2^{-1/x} = \infty \qquad \text{ and } \qquad \lim_{x \to 0^+} 2^{-1/x} = 0, </math>

which is true because <math display="inline">\lim_{x \to 0^-} {-1/x} = + \infty </math> and <math display="inline"> \lim_{x \to 0^+} {-1/x} = - \infty</math>

so that consequently,

<math display=block>\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</math>

whereas <math display="inline">\lim_{x \to 0^-} \frac{1}{1 + 2^{-1/x = 0</math> because the denominator diverges to infinity; that is, because <math>\lim_{x \to 0^-} 1 + 2^{-1/x} = \infty</math>. Since <math>\lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x)</math>, the limit <math>\lim_{x \to 0} f(x)</math> does not exist.

Relation to topological definition of limit

The one-sided limit to a point <math>p</math> 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 <math>p.</math>