1. The problem statement, all variables and given/known data The problem is to prove that the limit of sin(sqrt(x+1)) - sin(sqrt(x-1)) when x goes to infinity doesn't exist. 3. The attempt at a solution well, I converted sin(sqrt(x+1)) - sin(sqrt(x-1)) into the alternative form -2sin(sqrt(x+1)/2 - sqrt(x-1)/2)cos(sqrt(x+1)/2+sqrt(x-1)/2). Now my question is to show that limcos(x) as x goes to infinity doesn't exist. I tried to use epsilon-delta definition to prove it but I failed. I know that cos(x) is a periodic function and oscillates between 1 and -1 and I do know that its limit at infinity doesn't exist for that reason, but I want a mathematical proof that uses definitions or theorems to show that. Thanks in advance.