find the limit as x approaches 0 when (sin(1/x)+cos(1/x)) or prove/show it does not exist?

Question

Advanced Calculus limit problem and proof

Answer 1

The result of sine and cosine is always a number between -1 and 1, no matter how large the argument. So the sum of sin(1/x) and cos(1/x) will fluctuate between positive and negative values as x approaches zero and is therefore indeterminate.

cos(1/x)=sqrt(1-sin^2(1/x)), so if sin(1/x) is indeterminate, when x approaches zero, so is cos(1/x) despite the fact that sine and cosine of the same argument are related.