What exactly does the $\varepsilon$-$\delta$ definition of limits prove?

Question

e.g. Find the limit, $\lim\limits_{x \to 2} \ {\frac{2(x^2-4)}{x-2}}$, and prove it exists using the $\varepsilon$-$\delta$ definition of limits.

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This might be a stupid question, but I'm having a hard time wrapping my head around the $\varepsilon$-$\delta$ definition.

To my understanding, the limit exists if I'm able to find it, and the $\varepsilon$-$\delta$ proof requires that I already know the limit. So, what exactly does the $\varepsilon$-$\delta$ definition prove? Seems to me like it's about confirming the already-found-limit by showing the continuity of the limit's immediate surrounding. It seems unnecessary.

Answer 1

The $\varepsilon$-$\delta$ concept and related are essential for a rigorous definition of limits and also to prove the base cases (eg $1/x$, $x^n$, etc.) and all the theorems that we use for the calculation of limits in the more general cases. All foundamental theorems indeed requires $\varepsilon$-$\delta$ concept to be proved. Thus, even if we don't recognize it, we always are referring to this basic concept when we calculate a limit.

Note also that the limit concept is essential to define continuity in a rigorous manner thus, from a logical point of view, one comes before the other.

With reference to the example, this is a problem wich can be easily solved by continuity eliminating the common factor $(x-2)$ but consider that the aim of the exercise here is theoretical in order to become familiar with $\varepsilon$-$\delta$ definition.

Answer 2

That is not the purpose. You need a clear and precise definition of every mathematical concept. This is what it gives you. Once you have proved the basic theorems you do not need epsilon delta in practice. But to make progress in analysis the concept is essential.

Answer 3

I think your question is very nice and has some depth to it. Indeed, while we, as mathematicians, have the luxury to demand perfect rigor from our proof, we don't do it just to make ourselves feel intellectual.

While it is true that the $ε-δ$ definition doesn't tell us how to find the limit (which is perfectly reasonable as it just a definition not a theorem or a proposition) maybe you will appreciate it more with a "fun" activity: Give a definition of what continuity of a function means without the $ε-δ$ idea behind it.

I think the "problem" with t the $ε-δ$ definition is that in contrast with what we have dealt in high school, its precise usage and aim is clear much later after introducing it.

So don't be afraid that things aren't immediately clear, rather get used to it. With time (and practice) everything will make sense. The reason that excerises like "...use the $ε-δ$ definition to show that such limit is this" exist is only to make you more comfortable with the machinary and being able to understand more abstract concepts!

Answer 4

In a way you are right, in many cases we already know the limit by using higher-level computational rules (think of L'Hospital for example) that guarantee the existence of the limit at the same time.

The $\epsilon-\delta$ definition is the basic theoretical tool to prove a limit and it is used to establish the higher-level rules, such as "the limit of a product is the product of the limits, if they exist".

But there remain cases that you can' solve with the higher-level rules or you are unsure, then it is safe to revert to the basics.

Answer 5

Your doubts are genuine and perhaps they are a result of misinterpretation of definition of limit. The definition of limit is given in such a form that it can only be used to verify whether a number is a limit of some function at a point or not. It is not supposed to be used as a tool to find /evaluate limits. You can compare this situation with the fact that the definition of root of a polynomial hardly gives any practical method to find the root, but given a guess at the root one can use the definition to verify it.

Thus the standard approach in these problems is to guess the limit somehow and then use the definition to verify whether your guess is correct. How do I guess the limit then? Most problems give a specific number to be verified as a limit but this is not the case here and you need to guess the limit by using some calculation.

You may use your calculator and evaluate the function for values of $x$ near $2$ like for $x=2.1,2.01,2.001,1.9,1.99$ etc and find some pattern in the values of the function. If you are observant enough you will see that these values are all near the number $8$ and $8$ could be a good guess for the limit of the function under consideration. Now use the definition of limit to verify that $8$ is indeed the desired limit.

Also note that the above procedure of guessing and verifying a limit is practical only for very simple functions and giving such exercises for complicated functions is pointless. Using the definition of limit we can establish certain theorems (algebra of limits, squeeze, standard limits) which can be used to evaluate limits very efficiently. You should study the proofs of these theorems to understand the use of the definition of limit.