How do we know that lim f(n) = lim f(x), as a→∞ and x→∞, x∈R, n∈Z?

CogitoErgoCogitoSum @CogitoErgoCogitoSum

May 2021 4 71 Report

How do we know that lim f(n) = lim f(x), as a→∞ and x→∞, x∈R, n∈Z?

When we try to evaluate a limit, say for example lim n/e^n, we decide to treat the n not as an integer but as a real number, converting it into lim x/e^x. Now, because its a continuous variable, we can use l'Hôpital's Rule.

I am just thinking that converting from a discrete variable to a continuous variable is a hazard move. How can we arbitrarily do this? Are we not, in some implicit way, saying that infinity is an integer (which is nonsense)?

Update:

@Rocket. I didnt say infinity was a number. In fact, I EXPLICITLY said that its utterly nonsense. Why dont you read my question and UNDERSTAND IT before posting your stupid shiit.

Science & Mathematics / Mathematics

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The point you raise is quite valid. This move of introducing a real valued function of a real variable in order to use L'Hopital's rule (or some other theorem or methodology) has to be proven sound---it has.

Math's response about continuity having nothing to do with it is way off the mark. Sequence are functions and are (trivially) continuous, and L'Hopital's rule requires stronger conditions than continuity---specifically differentiability. But this isn't the issue.

One can state a formal theorem and provide proof to legitimize this practice. (Any decent calc text will include this.)

Thm: Suppose {a_n} is a sequence of real numbers and let f be a real valued function on IR such that for each positive integer n f(n) = a_n. Then lim(n->∞) a_n = L provided lim(x->∞) f(x) = L.

You can certainly allow for L to be one of +∞ or -∞ in addition to it being finite. It's not difficult to prove this.

Proof (L finite case): Let ε > 0 be given. As f(x) ->L we can choose M sufficiently large that for all x > M

|f(x) - L| < ε.

Then note that for all n > M

|a_n - L| = |f(n) - L| < ε.

Hence a_n -> L.

The proof for L infinite is similarly constructed.

I don't if you are taking classes or just studying on your own. It's fairly common to gloss over technical details in relatively low level classes. A introductory analysis text will fill in these gaps for you. We do this stuff all the time though---swapping the order of integration and summation, swapping log taking and limit taking, etc.---which is valid under the right circumstances; but we leave the formal details for another time.

Math

The fact that you say "we decide to treat the n not as an integer but as a real number" is quite worrying.

Integer is a real number itself.

It all depends on what f(x) is. L'hopital rule only applies to specific conditions. It does not apply in all cases. It has NOTHING to do with continuity, whatsoever. It is only used when both the numerator and denominator end up as 0/0 or infinity/infinity by direct substitution.

Getting lost in theoretical topics will not help. Practice actual problems. Sometimes direct substitution is all it takes to find a limit. Sometimes advanced techniques of laplace transform are needed.

It all depends on f(x) as I said. The idea of discrete to continuous makes no sense. Functions are never discrete. Discrete is only defined at integers. Functions are defined at the entire real domain, in most cases. There is no function that is discretely defined.

A discrete function would be f [x] but a continuous function would be f(x). Since we are dealing with parenthesis and not the brackets around the x, you are never to assume it is discrete

segarra

As x gets larger and larger, (e^-x) gets nearer and nearer to 0. on an identical time, (sin x) swings decrease from side to side between -a million and +a million. yet no count what the fee is for sin x, that's prolonged via a style that gets smaller and smaller. So the entire expression additionally gets smaller and smaller in the direction of 0. answer: 0.