Editing One-sided limit (section)

==Formal definition==

===Definition===
If <math>I</math> represents some [[Interval (mathematics)|interval]] that is contained in the [[Domain of a function|domain]] of <math>f</math> and if <math>a</math> is a point in <math>I</math> then the right-sided limit as <math>x</math> approaches <math>a</math> can be rigorously defined as the value <math>R</math> that satisfies:<ref>{{Cite book|last=Giv|first=Hossein Hosseini|url=https://books.google.com/books?id=Hf0mDQAAQBAJ&q=%22one-sided+limit%22|title=Mathematical Analysis and Its Inherent Nature|date=28 September 2016|publisher=American Mathematical Soc.|isbn=978-1-4704-2807-5|pages=130|language=en|access-date=7 August 2021}}</ref>{{Verify source|date=August 2021}}
<math display=block>\text{for all } \varepsilon > 0\;\text{ there exists some } \delta > 0 \;\text{ such that for all } x \in I, \text{ if } \;0 < x - a < \delta \text{ then } |f(x) - R| < \varepsilon,</math>
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:
<math display=block>\text{for all } \varepsilon > 0\;\text{ there exists some } \delta > 0 \;\text{ such that for all } x \in I,  \text{ if } \;0 < a - x < \delta \text{ then } |f(x) - L| < \varepsilon.</math>

We can represent the same thing 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>.

<!-- Right (positive) sided limit definition -->
:<math display=block>

\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))

</math>

<!-- Left (positive) sided limit definition -->
:<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 < a - x < \delta   \longrightarrow   | f(x) - L | < \varepsilon))

</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

<math display=block>|x - a| = |(-1)(-x + a)| = |(-1)(a - x)| = |(-1)||a - x| = |a - x|.</math>

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 [[Transitive relation|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.