Limits on cosine!

Calculus Level 3

$\large \lim_{n\to \infty} \cos \left ( \pi \sqrt{n^2+ n} \right)$

Calculate the limit above for integer $n$ .

The answer is 0.

1 solution

Haroun Meghaichi
Aug 26, 2014

Note that for $n>0$ : $\sqrt{n^2+n} = n\sqrt{1+\frac{1}{n} } = n\left(1+\frac{1}{2n} + o\left(\frac{1}{n}\right) \right)= n+\frac{1}{2} +o(1)$ The limit is clearly $\displaystyle \lim_{n\to \infty} \cos \left(n\pi +\frac{\pi}{2}+o(1)\right) =0$ .

Limits on cosine!

The answer is 0.

1 solution

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