# Formal Definition of Limit and Proofs

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.

• Intuition: If your $x$ is close to $x_0$ then $f(x)$ is close to $L$. And if you want to achieve a certain level of closeness for the latter, there exists a certain level of closeness for the former that can ensure that. Dec 11, 2013 at 16:02

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.

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.