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  • Evaluate    <img alt="\mathrm{\lim _{n \rightarrow \infty}(n+1)^{\frac{1}{\sqrt{n}}}}" src="https://entrancecorner.oncodecogs.com/gif.latex?%5Cmathrm%7B%5Clim%20_%7Bn%20%5Crightarrow

Evaluate    \mathrm{\lim _{n \rightarrow \infty}(n+1)^{\frac{1}{\sqrt{n}}}}

Option: 1

1


Option: 2

0


Option: 3

5


Option: 4

9


Answers (1)

best_answer

Note that
\mathrm{1 \leq(n+1)^{1 / \sqrt{n}} \leq(2 n)^{1 / \sqrt{n}}=2^{1 / \sqrt{n}}\left((\sqrt{n})^{1 / \sqrt{n}}\right)^2 }
If we take  \mathrm{2^{1 / x} }  and  \mathrm{x^{1 / x} \rightarrow 1 }  as  \mathrm{x \rightarrow \infty }  for granted, then
\mathrm{ 2^{1 / \sqrt{n}}\left((\sqrt{n})^{1 / \sqrt{n}}\right)^2 \rightarrow 1 \cdot 1^2=1 }
and the Squeeze Theorem does the rest.

Posted by

SANGALDEEP SINGH

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