TITLE: Find $\lim\limits_{n \rightarrow \infty}n\bigg(\cos \bigg(\frac {1}{\sqrt n} \bigg) - 1\bigg)$
QUESTION [2 upvotes]: $$\lim_{n \rightarrow \infty}n\bigg(\cos \bigg(\frac {1}{\sqrt n} \bigg) - 1\bigg)$$
I'm surprisingly struggling with this limit, could you give me a hint how to handle it (no L'Hospital and no prior knowledge what the limit value is)? 
Generally what are some basic methods to handle $\infty \cdot 0$?

REPLY [1 votes]: If you want to go beyond the limit itself, considering$$A=x\left(\cos \left(\frac {1}{\sqrt x} \right) - 1\right)$$ define $x=\frac 1 {t^2}$ making $$A=\frac 1 {t^2}\left(\cos \left(t \right) - 1\right)$$ Now, use the classical Taylor expansion 
$$\cos(t)=1-\frac{t^2}{2}+\frac{t^4}{24}+O\left(t^6\right)$$ which makes
$$A=-\frac{1}{2}+\frac{t^2}{24}+O\left(t^4\right)$$ which shows the limit when $t\to 0$ and also how it is approached.
If you want to go back to $n$, replace $t$ by $\frac 1 {\sqrt n}$ to get $$n\left(\cos \left(\frac {1}{\sqrt n} \right) - 1\right)=-\frac{1}{2}+\frac{1}{24n}+O\left(\frac 1 {n^2}\right)$$
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