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So far, we have defined the limit of a function $f$ as $x$ approaches $a$ qualitatively. In this section, we are going to introduce the definition of limit on a sound mathematical basis.
History
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Although mathematicians intuitively applied limiting processes even before the development of calculus, without a precise definition of limit, they were not able to prove important theorems of calculus with sufficient rigor. The first person who tried to put the definition on a mathematical sound basis was the French mathematician, engineer, and physicist, AugustinLouis Cauchy (1789–1857). Finally, the definitive modern definition of limit was formulated by the German mathematician Karl Weierstrass (1815–1897) who used two greek letters $\epsilon$ (epsilon) and $\delta$ (delta) for the small differences. For more information read this Wikipedia article.
The ????–???? Definition of Limit
Let’s review the informal definition of the limit of a function that says the limit of $f(x)$ as $x$ approaches $a$ is $L$ if we can make the values of $f(x)$ as close to $L$ as we please by taking $x$ sufficiently close to $a$, but not equal to $a$.
Because the inequality $f(x)L<\epsilon$ is equivalent to
\[\epsilon<f(x)L<\epsilon\]
or
\[L\epsilon<f(x)<L+\epsilon,\]
the geometrical meaning of this game is: you consider a band of width $2\epsilon$ bounded by the lines $y=L\epsilon$ and $y=L+\epsilon$ (see Figure 1), and I need to find an open interval of radius $\delta$ with $a$ at the center such that the entire points on the graph of $y=f(x)$ above the interval $(a\delta,a+\delta)$—except possibly the point above $a$ itself— lie within the band you gave me.

 Recall that an open interval with $a$ at the center is called a neighborhood of $a$. For more information see the Section on Absolute Value.
Figure 1 
For example, at the beginning of the Section on the Concept of a Limit, we saw that ${\displaystyle \lim_{x\to1}f(x)=4}$ where
\[f(x)=\frac{4x^{2}4}{2x2}.\]
For instance, if you give me $\epsilon=0.01$, I will take $\delta=0.005$ (or smaller), and claim $f(x)4<0.01$ for all $x\neq1$ satisfying $x1<0.005$; because if $x\neq1$ and $x1<0.005$ then
\begin{align*}
f(x)4 & =2x+24\\
& =2x2\\
& =2x1<2\times0.005=0.01.
\end{align*}
Recall that when $x\neq1$\[f(x)=\frac{4(x^{2}1)}{2(x1)}=2\frac{(x1)(x+1)}{(x1)}=2(x+1).\]
If you give me $\epsilon=0.0002$, I just need to take $\delta=0.0001$ (or smaller), because $x1<0.0001$ and $x\neq1$ implies that $f(x)4<0.0002$
\begin{align*}
f(x)4 & =2x+24\\
& =2x1<2\times0.0001=0.0002.
\end{align*}
If this game goes on for ever and for every $\epsilon>0$ you give me, I can find a $\delta>0$ with the aforementioned conditions, then we say the limit of $f(x)$ as $x$ approaches $a$ is $L$. Specifically, we state that if we can make $f(x)L$ less than any given positive number $\epsilon>0$ whenever $xa$ is less than some appropriately chosen positive number $\delta$ and $xa\neq0$ (because $x\neq a$) then
\[\lim_{x\to a}f(x)=L.\]
We remark that in general the size of $\delta$ depends on the size of $\epsilon$.
Instead of writing $xa<\delta$ and $xa\neq0$ (or $x\neq a$), we can concisely write \[0<xa<\delta.\]
Recall that the condition $x\neq a$ or $0<xa$ is imposed because we deal with the values of $f(x)$ for $x$ close to $a$ not equal to $a$, and the exact value of $f(x)$ at $x=a$ has no influence on the value or the existence of the limit.
Definition 1: Let $f$ be a function that is defined at every number in some open interval containing $a$ except possibly at the number $a$ itself. The symbol ${\displaystyle \lim_{x\to a}f(x)=L}$ means that for every $\epsilon>0$, however small, there exists a $\delta>0$ such that \[f(x)L<\epsilon\qquad\text{whenever}\qquad0<xa<\delta.\]
Another way of writing the last line is:
\[ \bbox[#F2F2F2,5px,border:2px solid black]{\large \text{“for all } x: \quad 0<xa<\delta\impliesf(x)L<\epsilon\text{ ”}}\]
We use the symbol “⟹” in place of “implies” or “if … then … .”
The above definition is often called the epsilondelta definition of limit.
Instead of saying “let $f$ be a function that is defined at every number in some open interval containing $a$ except possibly at the number $a$ itself”, we can say “let $f$ be defined in a deleted neighborhood of the point $a$.” For the definition of the deleted neighborhood, see the Section on Absolute Value.