Examples of Proofs of Limits

Question

Are there any good examples of the proof for limits? Every proof I see has examples like L=4 and things like that, but I'm trying to find something that shows the idea of limits is valid.

Thanks!

Answer 1

The question really should be "is it useful," because limit is a concept that is used to solve problems. Newton and Leibniz invented the concept of limits to study physics, so it is definitely useful. They did not create a careful definition, which only was created in the 19th century.

That is not to say that limits actually exist in the real world - who knows, the universe might be discrete. But even if discrete, the concepts of calculus - limits, continuity, etc., are really great for estimating. And there is no mathematics that is, so far, better than an estimator for the universe.

The Euclidean plane might never exist in the real world, but it is still really really useful for architecture when you don't have to worry too much about the curvature of the universe due to gravity or the curvature of the earth.

So, what does it mean for a definition to be "true?" And how do examples help verify that a definition is true? All examples can do is show that Newton and Leibniz's intuitive concept matches our definition in the cases where you can apply intuition, but does that mean it is "true?"

Answer 2

The formal definition is: $$\forall~\epsilon >0,~\exists~\delta > 0 :~ |x - a|<\delta \implies |fx - L|<\epsilon$$

The idea is that given a margin of error $\epsilon$, one can find a margin of tolerance for $x$, $\delta$, such that if $x$ is inside that margin ($|x - a| <\delta$), then $fx$ is satisfactorily close to the target ($|fx - L| < \epsilon$). Once I gave in an answer the general strategy for dealing with $\epsilon-\delta$ proofs for polynomials. Maybe you'll find it helpful. Also, I quite liked Leithold's approach to it.

Answer 3

I would choose a proof of the convergence of an infinite geometric series, such as in Wikipedia.

This links well to Zeno's paradox of Achilles and the Tortoise. Achilles runs a race with the tortoise, giving it a head start of 100 meters. Achilles runs this far in 10 seconds, but the tortoise has gone 10 meters. Achilles runs that quickly, but the tortoise has now gone an additional meter. When Achilles runs that distance, the tortoise has gone even farther. Zeno argued that the infinitely many distances that Achilles must cover shows that he never catches the tortoise and that hence motion is impossible.

Today we say that limits shows that Achilles does catch up. The distance that Achilles runs (in meters) is $$100 + 10 + 1 + \frac{1}{{10}} + \frac{1}{{100}} + \cdots = 111\frac{1}{9}$$

and the time it takes (in seconds) is $$10 + 1 + \frac{1}{{10}} + \frac{1}{{100}} + \cdots = 11\frac{1}{9}{\text{.}}$$

These limits show that Achilles does catch the tortoise in a finite time after running a finite distance.

Can it get more interesting than solving a 2000 year old problem?

Answer 4

Let's say you have the function $$f(x) = (x^2 - 1)/(x - 1)$$

Remember two functions are equal if and only if they have 1) they same values everywhere, and 2) they have the same domain. Well notice: $$(x^2 - 1)/(x - 1) =( (x + 1)(x - 1))/(x - 1) = (x + 1)$$

These functions are equal everywhere, with the exception that $f$ is not defined at x = 1. In this case, it doesn't make sense to say "what is f(1)?" Because 1 is not even in the domain of f.

The idea of limits is that we can get "as close as we want" to x = 1 in order to see what $f(x)$ approaches as x approaches 1.

Formally, we define this as follows:

$\lim_{x \to a} f(x) = L$ if for every $\delta > 0$, there exists some $\epsilon > 0$ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon$.

Geometrically, this means that if the limit exists, if we decide how small we want our $\epsilon$ to be, we can make the difference between $f(x)$ and $L$ less than $\epsilon$ by moving x close enough to $a$.