- [[prove the general limit exists at x]], [[double intersection]], [[limit laws]], [[continuity]]
# Idea
The limit of the function $f$ as $x$ approaches $a$ is the number $L$ if, as $x$ gets closer and closer to $a$, the function values $f(x)$ get closer and closer to $L$. Note that we never concern ourselves with the value of $f(x)$ when $x = a$.
For function $f(x)$, as $x$ approaches $a$, the limit is $L$.
$\lim _{x \rightarrow a} f(x)=L$
This limit is also known as the **overall limit** (in contrast to left-hand and right-hand limit).
If there is no such number $L$, we say the limit does not exist.
$\lim _{x \rightarrow c} f(x)$ has nothing to do with the value of $f$ at $a$ itself. It only says something about what happens as gets close to $a$.
For the precise definition of limit, see [[limit - epsilon delta definition]].
Limits only exist for [[continuous functions]], which are well-behaved.
## A limit is like an unattainable goal
> A limit is like an unattainable goal. You can get closer to it, but you can never get all the way there. - [[Strogatz 2019 infinite powers - how calculus reveals the secrets of the universe]]
Approaching a value as close as desired without necessarily reaching it, or "moving closer and closer" but never at a particular value: Note that the limit does not depend on the value of the function $f$ at the point of interest. When we say $x$ is approaching 1, we're not concerned with the value $x = 1$ itself, but only when $x$ is near or approaching 1.
That is, we ask the following: as $x$ move closer and closer to some number, does $f(x)$ also move closer to something?
## Approaching from the left versus right: Left and right-hand limits
![[s5EjF9JwlP-c1q2p1.gif]]
$
f(x) \rightarrow L \text { as } x \rightarrow a^{-}
$
$
\lim _{x \rightarrow \mathbf{a}^{-}} f(x)=L
$
$
f(x) \rightarrow R \text { as } x \rightarrow a^{+}
$
$
\lim _{x \rightarrow \mathbf{a}^{+}} f(x)=R
$
The function below has a jump discontinuity at $a$. This means that if an ant were to crawl along the curve, it would have to jump at $x=a$ to get from one piece to the other. It means that $\lim _{x \rightarrow a} f(x)$ does not exist.
$\lim _{x \rightarrow a} f(x)=L$ exists if and only if $\lim _{x \rightarrow a^{+}} f(x)=L$ and $\lim _{x \rightarrow a^{-}} f(x)=L$. That is, the limit $\lim _{x \rightarrow a} f(x)=L$ exists only if the left and right hand limits equal $L$.
If a limit does not exist, we write $DNE$.
![[20240317135600.png]]
![[tumblr_6cdf62959b18340729980dca48deb1ce_904c3c12_1280.gif]]
## Examples
### Approaching a value $x$ from the left versus right
Note that $x = 1$ is undefined for the function below because of the denominator $x - 1$.
$
f(x)=\frac{\sqrt{3-5 x+x^2+x^3}}{x-1}
$
As $x$ approaches 1 from the left ($x \rightarrow 1^{-}$), $f(x)$ approaches -2.
- $x \rightarrow 1^{-}$, $f(x) \rightarrow -2$
- left-hand limit
$
\lim _{x \rightarrow 1^{-}} f(x)=-2
$
As $x$ approaches 1 from the right, $f(x)$ approaches 2.
- $x \rightarrow 1^{+}$, $f(x) \rightarrow 2$
- right-hand limit
$
\lim _{x \rightarrow 1^{+}} f(x)=2
$
```functionplot
---
bounds: [-2, 3, -3, 3]
xLabel: x
yLabel: f(x)
disableZoom: true
---
f(x) = (3 - 5x + x^2 + x^3)^0.5 / (x - 1)
```
### Discontinuous functions
![[20240315224851.png]]
$
\lim _{x \rightarrow 1} f(x) = 1
$
As the function gets closer and closer to $x=1$, the value of the function approaches $1$, so this is the value of the limit, even though the value of the function at $1$ is something totally different!
![[20240315225201.png]]
The limit does not exist. From the left, the function is approaching −1. From the right, the function is approaching 1. Since these are different values, the limit does not exist.
![[20240319231113.png]]
$\lim _{x \rightarrow(-2)^{-}} f(x)=3$
$\lim _{x \rightarrow(-2)^{+}} f(x)=1$
$\lim _{x \rightarrow(-2)} f(x)=\mathrm{DNE}$
$f(-2)=2$
# References
- [3Blue1Brown - Limits and the definition of derivatives](https://www.3blue1brown.com/lessons/limits)
- [Precise Definition of a Limit](https://www.sfu.ca/math-coursenotes/Math%20157%20Course%20Notes/sec_LimitsFormal.html)
- [The precise definition of the limit EXPLAINED! (KristaKingMath) - YouTube](https://www.youtube.com/watch?v=rSxxEtKVw-w)
- [precise definition of the limit](https://courses.kristakingmath.com/library/limits-continuity-44602/110493/path/step/113604895/)
- [Practice Calculus Fundamentals | Brilliant](https://brilliant.org/courses/calculus-done-right/introduction-107/function-limits/1/?from_llp=advanced-math)
- [Course | MITx Online](https://courses.mitxonline.mit.edu/learn/course/course-v1:MITxT+18.01.1x+2T2023/block-v1:MITxT+18.01.1x+2T2023+type@sequential+block@lim_1-sequential/block-v1:MITxT+18.01.1x+2T2023+type@vertical+block@lim_1-tab7)
- [Course | MITx Online](https://courses.mitxonline.mit.edu/learn/course/course-v1:MITxT+18.01.1x+2T2023/block-v1:MITxT+18.01.1x+2T2023+type@sequential+block@lim_1-sequential/block-v1:MITxT+18.01.1x+2T2023+type@vertical+block@lim_1-tab11)