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cmowla

Member

Registered: 2012-06-14

Posts: 70

Easier Proofs for the Basic Limit Laws?

I've taken all basic Calculus courses and Advanced Calculus I (not that I learned any additional information on proving the basic limit laws in that class...), but I still do not know if what is in my textbook (or what I have researched on the net) is the easiest (intuitive) way to prove the basic limit laws (sum , product, reciprocal, and constant).

I came up with arguments that I highly doubt are valid proofs, since I have never seen an explanation this simple in a proof of these laws by those who are much more mathematically sound than I am.

The main theme of the "arguments" for each of the basic limit laws is the following statement.

Since

is an arbitrary number, then

is an arbitrary number

Now before I show the arguments, I make note of the formal definition of the (two sided) limit.

I evaluated the absolute value to instead have:

Since this adjusted formal definition shows that the definition can be broken into two parts (we have two statements because of the or operator), we can just handle the first part

, and we can understand that the argument is similar if we were to tackle the second part of the definition.

Here are the arguments/"proofs" using this theme.  To save space, let (*) represent the statement in the first quote (epsilon times a number is an arbitrary number/another epsilon..., etc.)

Sum Law

Argument

Constant Law

Argument

Product Law

Argument

Reciprocal Law

Argument

We can of course prove the difference law with the results of the sum and constant laws, and we can prove the quotient law using the results of the product and reciprocal laws.

Is (*) (the statement in the first quote) a correct assumption?  If so, then are these arguments valid proofs?

benice

Member

Registered: 2010-06-10

Posts: 117

Website

Re: Easier Proofs for the Basic Limit Laws?

is an arbitrary number, then

is an arbitrary number

Is (*) (the statement in the first quote) a correct assumption?

It is correct (but informal) since the following two statements are equivalent.

If so, then are these arguments valid proofs?

cmowla

Member

Registered: 2012-06-14

Posts: 70

Re: Easier Proofs for the Basic Limit Laws?

It is correct (but informal) since the following two statements are equivalent.

Thanks.  I love that version of the formal definition!

If so, then are these arguments valid proofs?

Sorry that I implied that they themselves are "stand alone" arguments, but in the proof I have seen for the product of the limits, for example, the only appearances of delta is to restate the restriction of the formal definition (the relationship between chosen epsilons and their corresponding deltas), but I think epsilons are just used in the actual arithmetic.

I stated the formal definition at the beginning of that post to state it.  So, if these arguments are correct, then I just could have added the relationship between the chosen epsilon and the corresponding delta...is that all you were implying?  (As I have said, I apologize for not mentioning that the arguments are not stand alone ones, i.e., the "background information" is provided for all arguments at the beginning of that post).

I still am not sure if these arguments are valid proofs (given now that I explained that I "abbreviated" them by providing background information for all arguments beforehand).

Last edited by cmowla (2012-07-02 07:15:48)

Ivar Sand

Member

Registered: 2013-01-22

Posts: 16

Re: Easier Proofs for the Basic Limit Laws?

In your post #1 you state that you evaluated the absolute value(s) in the second quote to produce the statement in the third quote. However, I have a counterexample. Set x = a + δ/2 and f(a + δ/2) = L - 2ε. In this case the statement in the second quote is false, whereas the statement in the third quote is true.

Also, in the proof of Sum Law, the theorem statement has ε whereas the last statement in the proof has ε_2. Shouldn't these two epsilons have been the same? The same or something similar goes for Constant Law and Product Law.

Last edited by Ivar Sand (2023-09-13 01:36:35)

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I majored in Physics in 1976. Also, I studied mathematics and computer science. I worked as a computer programmer. I became a pensioner in 2016. I am from Norway.