Different approaches to a limit problem

Question

Evaluate :

Definite Integral

The standard way is to rewrite the given limit in the form of a definite integral:

= 0

Complex Partners

Let

\ C = 0

No sweat

An easy way to show this depends on the knowledge of trigonometric function, without even taking the limit:

(1) is changed to (2) by making use of the fact:

Add (1) and (2) and divide by 2, we have:

Final challenge

Does the limit:

exists? If yes, what is its value?

Solution

1. Using integral

2. Using complex number (as in the above):