Limits Question

Question

lim n tends to infinity

((cosx)ⁿ + (sinx)ⁿ)^1/n

^{for x belongs to (0,π/4)}

Answer 1

Multiply the expression by 1 in the form (cos x)/(cosⁿx)^1/n

The limit then becomes lim (as n -> infinity) cos x(1+ tanⁿx)^1/n

Since 0<tan x<1 on the specified interval, the term in parentheses is between 1 and 2 and, therefore, goes to 1 as n increases. This means the limit is cos x. I am not sure I can give an epsilon/delta derivation for this limit, but I believe it is correct. If you get a different answer in class, I would appreciate knowing about it. Thanks!