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.