Limits and derivatives

Question

Limit n-> infinity ( sin^n 1 + cos ^n 1 ) ^n

Answer 1

For any n > 1 the sum of the sine and cosine terms must be < 1. This is a consequence of the fact that max value of either is < 1. So raising to any power makes each terms even smaller. Adding them together will still be less than one and raising the sum to the nth power makes it even smaller. So in the limit the expression becomes zero.

Answer 2

We know that sin²(1) + cos²(1) = 1

Since sin(1) and cos(1) are both less than 1,

sinⁿ(1) + cosⁿ(1) < 1 for n > 2

That means that the limit as n -> ∞ = 0