To understand what limits are, let's look at an example. We start with the function $f(x)=x+2$.
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The limit of $f$ at $x=3$ is the value $f$ approaches as we get closer and closer to $x=3$. Graphically, this is the $y$-value we approach when we look at the graph of $f$ and get closer and closer to the point on the graph where $x=3$.
For example, if we start at the point $(1,3)$ and move on the graph until we get really close to $x=3$, then our $y$-value (i.e. the function's value) gets really close to $5$.
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Similarly, if we start at $(5,7)$ and move to the left until we get really close to $x=3$, the $y$-value again will be really close to $5$.
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For these reasons we say that the limit of $f$ at $x=3$ is $5$.
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You might be asking yourselves what's the difference between the limit of $f$ at $x=3$ and the value of $f$ at $x=3$, i.e. $f(3)$.
So yes, the limit of $f(x)=x+2$ at $x=3$ is equal to $f(3)$, but this isn't always the case. To understand this, let's look at function $g$. This function is the same as $f$ in every way except that it's undefined at $x=3$.
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[[☃ explanation 2]]
Just like $f$, the limit of $g$ at $x=3$ is $5$. That's because we can still get very very close to $x=3$ and the function's values will get very very close to $5$.
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So the limit of $g$ at $x=3$ is equal to $5$, but the value of $g$ at $x=3$ is undefined! They are not the same!
That's the beauty of limits: they don't depend on the actual value of the function at the limit. They describe how the function behaves when it gets close to the limit.
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We also have a special notation to talk about limits. This is how we would write the limit of $f$ as $x$ approaches $3$:
$\begin{align}
\scriptsize\text{"The limit of..."}&\qquad\scriptsize\text{"...the function }f\text{..."}
\
\searrow\qquad&\qquad\swarrow
\
\LARGE\displaystyle\lim_{x\to 3}&\LARGE f(x)
\
\nearrow\qquad
\
\scriptsize\text{"...as }x\text{ approaches }3\text{."}
\end{align}$
The symbol $\displaystyle\lim$ means we're taking a limit of something.
The expression to the right of $\displaystyle\lim$ is the expression we're taking the limit of. In our case, that's the function $f$.
The expression $x\to 3$ that comes below $\displaystyle\lim$ means that we take the limit of $f$ as values of $x$ approach $3$.
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In limits, we want to get infinitely close.
What do we mean when we say "infinitely close"? Let's take a look at the values of $f(x)=x+2$ as the $x$-values get very close to $3$. (Remember: since we're dealing with limits we don't care about $f(3)$ itself.)
| $x$ | $f(x)$ |
|---|---|
| $2.9$ | $4.9$ |
| $2.99$ | $4.99$ |
| $\gray{\underbrace{\color{black}{2.999}}_{\text{close to }3}}$ | $\gray{\underbrace{\color{black}{4.999}}_{\text{close to }5}}$ |
We can see how, when the $x$-values are smaller than $3$ but become closer and closer to it, the values of $f$ become closer and closer to $5$.
| $x$ | $f(x)$ |
|---|---|
| $3.1$ | $5.1$ |
| $3.01$ | $5.01$ |
| $\gray{\underbrace{\color{black}{3.001}}_{\text{close to }3}}$ | $\gray{\underbrace{\color{black}{5.001}}_{\text{close to }5}}$ |
We can also see how, when the $x$-values are larger than $3$ but become closer and closer to it, the values of $f$ become closer and closer to $5$.
Notice that the closest we got to $5$ was with $f(2.999)=4.999$ and $f(3.001)=5.001$, which are $0.001$ units away from $5$.
We can get closer than that if we want. For example, suppose we wanted to be $0.00001$ units from $5$, then we would pick $x=3.00001$ and then $f(3.00001)=5.00001$.
This is endless. We can always get closer to $5$. But that's exactly what "infinitely close" is all about! Since being "infinitely close" isn't possible in reality, what we mean by $\displaystyle\lim_{x\to 3}f(x)=5$ is that no matter how close we want to get to $5$, there's an $x$-value very close to $3$ that will get us there.
If you find this hard to grasp, maybe this will help: how do we know there are infinite different integers? It's not like we've counted them all and got to infinity. We know they are infinite because for any integer there's another integer that's even larger than that. There's always another one, and another one.
In limits, we don't want to get infinitely big, but infinitely close. When we say $\displaystyle\lim_{x\to 3}f(x)=5$, we mean we can always get closer and closer to $5$.
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Another example: $\displaystyle\lim_{x\to 2}x^2$
Let's analyze $\displaystyle\lim_{x\to 2}x^2$, which is the limit of the expression $x^2$ when $x$ approaches $2$.
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We can see how, when we approach the point where $x=2$ on the graph, the $y$-values are getting closer and closer to $4$.
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We can also look at a table of values:
| $x$ | $x^2$ |
|---|---|
| $1.9$ | $3.61$ |
| $1.99$ | $3.9601$ |
| $\gray{\underbrace{\color{black}{1.999}}_{\text{close to }2}}$ | $\gray{\underbrace{\color{black}{3.996001}}_{\text{close to }4}}$ |
| $x$ | $x^2$ |
|---|---|
| $2.1$ | $4.41$ |
| $2.01$ | $4.0401$ |
| $\gray{\underbrace{\color{black}{2.001}}_{\text{close to }2}}$ | $\gray{\underbrace{\color{black}{4.004001}}_{\text{close to }4}}$ |
We can also see how we can get as close as we want to $4$. Suppose we want to be less than $0.001$ units from $4$. Which $x$-value close to $x=2$ can we choose?
Let's try $x=2.001$:
$2.001^2=4.004001$
That's more than $0.001$ units away from $4$. Alright, so let's try $x=2.0001$:
$2.0001^2=4.00040001$
That's close enough! By trying $x$-values that are closer and closer to $x=2$, we can get even closer to $4$.
In conclusion, $\displaystyle\lim_{x\to 2}x^2=4$.
A limit must be the same from both sides.
Coming back to $f(x)=x+2$ and $\displaystyle\lim_{x\to 3}f(x)$, we can see how $5$ is approached whether the $x$-values increase towards $3$ (this is called "approaching from the left") or whether they decrease towards $3$ (this is called "approaching from the right").
[[☃ image 1]]
Now take, for example, function $h$. The $y$-value we approach as the $x$-values approach $x=3$ depends on whether we do this from the left or from the right.
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When we approach $x=3$ from the left, the function approaches $4$. When we approach $x=3$ from the right, the function approaches $6$.
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When a limit doesn't approach the same value from both sides, we say that the limit doesn't exist.
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