Calculate the following limit: lim
I have tried to use the squeeze theorem and other convergence tests but all failed.
Please, any help?
Answer
Note that
1\le(n+1)^{1/\sqrt n}\le(2n)^{1/\sqrt n}=2^{1/\sqrt n}((\sqrt n)^{1/\sqrt n})^2
If we take 2^{1/x} and x^{1/x}\to1 as x\to\infty for granted, then
2^{1/\sqrt n}((\sqrt n)^{1/\sqrt n})^2\to1\cdot1^2=1
and the Squeeze Theorem does the rest.
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