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