Positive Integer for finding a Limit

[yojoe]

I have a good one for you:

Find the positive integer k for which lim x-->0 (sin(sin(x))/x^k exists, and then find the value of this limit.


I first started out with zero for k, which gave me a limit of zero. Then I used 1 for k, which gave limiting values close to zero. Then all consecutive numbers used after that started giving me increasing exponential values.

Which one is correct for the integer? 0 or 1? And are my limits correct for either number?

Thanks!

 

[Matt]

I first started out with zero for k...

But 0 is not positive.

Then I used 1 for k, which gave limiting values close to zero.

When k=1, the limit is 1 by L'Hospital's Rule. For any other value of k, the limit is infinity.