It helps to think about the core concepts of Calculus from a young age, if you want to develop your expertise or talents further in math, pure or applied, engineering or mathematical sciences. At a tangible level, it helps you attack more or many questions of the IIT JEE Advanced Mathematics. Let us see if you like the following proof, or can absorb/digest it:
Reference: Calculus and Analytic Geometry by Thomas and Finney, 9th edition.
The Sandwich Theorem:
Suppose that for all x in some open interval containing c, except possibly at
itself. Suppose also that
. Then,
.
Proof for Right Hand Limits:
Suppose . Then, for any
, there exists a
such that for all x, the inequality
implies
and
….call this (I)
These inequalities combine with the inequality to give
….call this (II)
Therefore, for all x, the inequality implies
. …call this (III)
Proof for LeftHand Limits:
Suppose . Then, for
there exists a
such that for all x, the inequality
implies
and
…call this (IV).
We conclude as before that for all x, implies
.
Proof for Two sided Limits:
If , then
and
both approach L as
and as
so
and
. Hence,
. QED.
Let me know your feedback on such stuff,
Nalin Pithwa