Formal Definition of Limit and Proofs

Question

I'm having trouble understanding the formal definition of a limit...

Let $f(x)$ be defined on an open interval about $x_0$, except possibly at $x_0$ itself. We say that the limit of $f(x)$ as $x$ approaches $x_0$ is the number $L$, and write

$\lim\limits_{x \to x_0} f(x) = L$,

if, for every number $\epsilon > 0$, there exists a corresponding number $\delta > 0$ such for all $x$,

$0 < |x - x_0| < \delta => |f(x) - L| < \epsilon$.

_{From Thomas' Calculus 12th Edition, page 58}

This definition is then used to prove that $\lim \limits_{x \to x_0} x = x_0$ and $\lim \limits_{x \to x_0} k = k$.

Can anyone give me a little bit of intuition behind the definition and these proofs? I'm really having trouble understanding them.

Answer 1

It may help to think of $\epsilon$ as the error you're willing to tolerate between $f$ and $L$. Then $\delta$ is the deviation which is allowed for $x$ from $x_0$ to get the error less than $\epsilon$.

The definition is then simply saying that no matter how small you wish your error to be, there is some distance around $x_0$, the deviation, which will allow you to attain an error less than your bound.

Answer 2

The definition is quite subtle, but I will explain exactly what

$\lim_{x\to{x_0}} f(x) $

means.

Def. Let $f(x)$ be function defined for all $ x \neq x_0$, letting $L$ be a real number, we have hence:

$\lim_{x\to{x_0}} f(x) = L$

if, for every $\varepsilon > 0$, there is a $\delta > 0$, such that if $0 < |x- x_0|< \delta$, hence $|f(x) - L| < \varepsilon$.

Now, we will make everything clear. What the definition says is: if you pick an $ \varepsilon > 0$ (fixes it!), a positive number in the vertical $y$ axis, then naturally, if you draw a horizontal line in between $L+\varepsilon$ and $L-\varepsilon$, (for a prefixed L obviously), if you touch this line with the $f(x)$, you are going to have a point of intersection, with coordinates $(\delta, \varepsilon)$, however $\delta$ varies, in the sense that you can have each time, smaller intervalls within the intervall $\delta + x_0$ and $ x_0 - \delta$, so you can pick another number $\delta_0$ such that the intervall $(x_0 - \delta_0, x_0 + \delta_0)$ is "inside" the interval $(x_0 - \delta, x_0 + \delta)$.

Now, this $\delta$ exists, and it depends on $\varepsilon$, and it's clear that if x is in a distance to $x_0$ smaller than $\delta$, if we repeat the process of "line association" we've done, then, we'll obtain naturally y's in the y axis closer to $L$ than $\varepsilon$ the distance we have started with. I think that examples are important, but in this case it can diverge, however if we think a bit beyond, we can do the following:

1. Pick $\varepsilon > 0$
2. Pick $\delta = \frac{\varepsilon}{k*}$, (which is clearly smaller than $\varepsilon$!), in such a way that when we put in the special property we get the $|x - x_0| < \varepsilon$. This pick a $\delta$ isn't arbitrary, so you have to manipulate the limit in order to find an $\varepsilon$; the thing to keep in mind is to make $|x - x_0|$ appears somewhere.
3. Then naturally it's clear that the definition we gave will hold.

*k is a number greater equal to 1.