The definition of "limit as x goes to infinity" is: if and only if, given any there exist X such that if x> X then . That is impossible for cos(px) because, no matter how large x is, there exist values of x ( for n an even integer) such that cos(x)= 1 and values of x ( for n an odd integer) such that cos(x)= -1. Since it keeps going from -1 to 1, it can't converge to a specific limit. The limit does not exist.

Solve what problem? As I said, limit of cos(px), as x goes to infinity, does not exist. If you are asking about that formula for the Dirac Delta function, the Dirac Delta Function is a "distribution" or "generalized function" , NOT a function. In any case, I don't see how that limit has anything to do with the delta function which can be described as "infinite if p= 0, 0 for all other values of p" which is not the case for that limit. Where did you get that formula?From the Dirac delta function, del(p) = limit(x tends to infinite) {sin(px)/(pi*p)}.

here 'p' is a constant.

Instead of using the relation between the sin and cos, Is there any other way to solve the above problem.

Use "LaTex". On this board begin a mathematical expression with .another request : How to type mathematical expression in the threads?

Here is a good tutorial on LaTex:

Getting to Grips with Latex - Mathematics - Latex Tutorials by Andrew Roberts @ School of Computing, University of Leeds

Note that different boards may use different html tags to start and end LaTex. I have seen "" (here), "", and even "\( \)".