what is the limit of x sin(1/x) as x approches 0?

Answer 1

Look at what happens to the parts.

x ==> 0 as x ==> 0 but

sin(1/x) does not converge to anything.

However, it is bounded by [-1,1]. Therefore, the broduct of those two functions will be bounded by [-x, x]. It is easy to show that this limit is zero. Pick any value of delta. As long as you choose x = epsilon, where epsilon is less than delta, you are assured that the function value is within delta of zero.

So -- the limit is zero.

Answer 2

Sin(angle) varies from +1 to -1. This can be ignored in the limit. So the problem reduces to limit x as x tends to 0.
This is obviously 0.