*We continue our presentation of basic stuff from Calculus and Analytic Geometry, G B Thomas and Finney, Ninth Edition. My express purpose in presenting these few proofs is to emphasize that Calculus, is not just a recipe of calculation techniques. Or, even, a bit further, math is not just about calculation. I have a feeling that such thinking nurtured/developed at a young age, (while preparing for IITJEE Math, for example) makes one razor sharp.*

We verify a few famous limits.

**Formula 1:**

If ,

We need to show that to each there corresponds an integer N so large that for all n greater than N. Since , while . there exists an integer N for which . In other words,

. Call this (I).

This is the integer we seek because, if , then

for all . Call this (II).

Combining I and II produces for all , concluding the proof.

**Formula II:**

For any number x, .

Let . Then, ,

as we can see by the following application of l’Hopital’s rule, in which we differentiate with respect to n:

, which in turn equals

.

Now, let us apply **the following theorem** with to the above:

(a theorem for calculating limits of sequences)** the continuous ****function theorem for sequences:**

Let be a sequence of real numbers. If be a sequence of real numbers. If and if f is a function that is continu0us at L and defined at all , then .

*So, in this particular proof, we get the following:*

.

**Formula 3:**

For any number x,

Since ,

all we need to show is that . We can then apply the Sandwich Theorem for Sequences (Let , and be sequences of real numbers. if holds for all n beyond some index N, and if ,, then also) to conclude that .

The first step in showing that is to choose an integer , so that . Now, let us the rule (formula 1, mentioned above), so we conclude that:. We then restrict our attention to values of . For these values of n, we can write:

, where there are factors in the expression , and

the RHS in the above expression is . Thus,

. Now, the constant does not change as n increases. Thus, the Sandwich theorem tells us that because .

That’s all, folks !!

Aufwiedersehen,

Nalin Pithwa.

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