I had been taught the formal definition of a limit with quantifiers. For me, it is very hard to follow and I understand very little of it. I was told that:
$$\text{If} \ \lim\limits_{x\to a}f(x)=L, \ \text{then:}$$
$$\forall \epsilon, \ (\epsilon > 0) \implies \exists \delta \ (\delta > 0 \ \text{and} \ \forall x, \ ((x\neq a \ \text{and} \ |x-a| < \delta) \implies |f(x)-L| < \epsilon))$$
I have absolutely no idea what this means. I think I get the first part, which is:
"For all $\epsilon$, if $\epsilon > 0$, then there exists a $\delta$ such that $\delta > 0$…"
From that point on I do not get it. I am not sure if what I wrote above is even correct. I am worried that this may be a very important definition to understand and memorize, so I need your help understanding it. A few questions that I have are:
$1$. Where did the $\epsilon$ and the $\delta$ even come from?
$2$. Why does $\epsilon$ and $\delta > 0$?
$3$. What are the absolute value signs for?
I would greatly appreciate some help, thanks!
Best Answer
Seth's answer here does more than enough to expose the intuition behind the definition of a limit. But I get the feeling you are having trouble untangling the definition itself. So I think this could be useful. Here is what it says:
"Forany given $\epsilon \gt 0$, however small, there is (and there always is) a corresponding positive quantity $\delta$ such that the value of the function $f(x)$ is less than $\epsilon$ distant from $L$ whenever the $x$ values are less than $\delta$ distant from $a$".
No matter how close you want your function to be to $L$ there is what we call a "neigbourhood" made of all points whose distance from $a$ is $\delta$ such that the value of the function $f(x)$ is as close as you previously wanted to $L$ for every value of $x$ in the so stipulated $\delta$ - neighbourhood.
If this is true then we say $L$ is the limit of the function $f$ as $x$ tends to (approaches) $a$.
All of this obviously is given in mechanical terms above. You have untangled it to a certain extent CORRECTLY. There onwards,
The $\forall x$ to me is superfluous. What you have inside the brackets is $|x - a| \lt \delta \implies |f(x) - L| \lt \epsilon$ which essentially means If $|x - a| \lt \delta $ then $|f(x) - L| \lt \epsilon$ or $|f(x) - L| \lt \epsilon \;\;$ whenever $|x - a| \lt \delta$ as I have explained above.
Hope I helped.