Finding xn as n Tends to Infinity: Q1 & Q2

[Mattofix]

Homework Statement

2 Questions, both find xn as n tends to infinity.

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Homework Equations

The Attempt at a Solution

Have attempted question one but am unsure if (1/n)log(n^2) tends to 0, and if it does do i need to prove it? I don't know how to do the second q, i know that sin(expn) oscillates between -1 and 1 and exp(-n) tends to 0 as n tends to infinity

 

[HallsofIvy]

Yes, (1/n) log(n^2) = (2/n)log(n) goes to 0. You might prove that by looking at 2ln(x)/x^2 and using L'Hopital's rule.

As for the second one, since sin is always between -1 and 1, you really just need to show that $\sqrt{n}/(n+ e^{-n})< \sqrt{n}/n$ (since $e^{-n}$ is always positive) converges to 0.

 

[Mattofix]

thanks