proving that lim of x as x approaches 0 is not equal to 0

Oh ! I didn't see Ian Fowler's answer before to write mine. Sorry !

Thanks , but I was thinking if the perimeter of shape (polygon ) be equal to circumference or will it APPROACH circumference . More importantly are approach and equal the same as when we evaluate any limit we in end say that limit IS EQUAL TO some value , or there is no limit, but how can something APPROACHING EQUAL TO SOMETHING . When we evaluate limit as x approaches zero , it is very similar to f(0), so why do we need limit st all.

@YAGYA GOYAL WISDOM I will try my best. Because in some cases f(0) does not exists or is indeterminate.

There is a subtle but important difference between these 2 statements:

1) As x --> 0 the expression f(x) = sin(x)/x --> 1 but never takes on the value 1.

2) lim(as x -->0)[sin(x)/x] = 1 and equality is achieved. But it is not equal = f(0) since f(0) is indeterminate. The graph of f(x) = sin(x)/x has a hole at x = 0 even though the limit exists.

i.e. The ---> 1 means we can make sin(x)/x arbitrarily close to 1 by making x sufficiently close to 0 and 1 is the only number that has this property. And "in the limit" equality is achieved. So this answers your question: so why do we need limit st all

As far as the infinite series 1/2 + 1/4 + 1/8 + ... is concerned the "..." means we are adding an infinite number of terms and equality is achieved and the infinite sum actually equals 1.

To say that 1/2 + 1/4 + 1/8 + .... approaches 1 is incorrect.

There is no difference between: sum(from n=1 to inf) [1/2^n] = 1 and 1/2 + 1/4 + 1/8 + ... = 1

The same is true for 9.99999.... = 10. This is exactly the same situation as the above. It is an infinite geometric series which has a definite finite value.

That's the best I can do.

@Phinehas That's quite an accusation. In 40 years I have never been accused of that by either professors, colleagues or students. I could accuse both of you and (@YAGYA GOYAL WISDOM) of the same thing but I will defer that judgement to the readers at large. Before you respond I highly suggest that you both watch the Mathologer video:

https://www.youtube.com/watch?v=SDtFBSjNmm0

Burkard really knows his stuff and I have seen it his way for decades. I think you both should listen and learn. Bown/Blue was throwing out some wild ideas and metrics for us to consider but never once did he suggest anything that contradicts what is to follow.

The crux of the problem seems to be whether you can actually legitimately assign a finite value to an infinite sum, and if you can, what is a reasonable way to go about assigning such a value. The " ...." assumes an infinite sum so:

sum(n=0--->inf)[1/2^n] = 2 is equivalent to 1 + 1/2 + 1/4 + 1/8 + ... = 2

If you look at the sequence of partial sums:

1, 3/2, 7/4, 15/8, 31/16, 63/32, .... the limit this sequence as n -->inf is 2. Therefore the only reasonable value to assign the infinite series is the value of 2. We are not talking about physically actually adding up an infinite number of terms on a calculator or by any other means but the abstract idea of considering the result of adding up an infinite number of terms. If you add a finite number of terms, no matter how large, then you achieve a value arbitrarily close to 2 but not actually equal to 2. However, an infinite number of terms actually equals 2 - exactly. The "..." does not mean a large finite number but an infinite number of terms. It's really not that hard to accept.

The same argument is made when trying to understand 9.9 repeat. So 9.9 repeat and 10 represent the same, I repeat, the same point on the real number line and therefore represent the same real number. See Mathologer.

I think this conversation should come to a close.

Cheers - Ian