Does the limit of cos√(2004+x) - cos√x exist?

[KLscilevothma]

This question appeared in today's exam but I think I did it wrongly. I use the "sum to product formula", but I can't find a limit value and say it doesn't exist. Am I correct?

$$\lim_{n\rightarrow\infty} cox \sqrt{2004 + x} - cos \sqrt{x}$$

 

[arildno]

I take it that you are to find the limit when x->inf, not n->inf.
The limit is zero:
Note that sqrt(a+x)=sqrt(x)*sqrt(1+(a/x)) approx. sqrt(x)+1/2(a/sqrt(x)), when a<<x

Hence, we may write the original cosine as cos(sqrt(x)+e),
where e->0 as x->inf.

Using sum-to-product, we have to evaluate the limit of:
cos(sqrt(x))*(cos(e)-1)-sin(e)*sin(sqrt(x)).

Since cos(sqrt(x)), sin(sqrt(x)) are bounded by 1, we see that the whole expression goes to 0.

 

[KLscilevothma]

I see. Thanks