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In my maths lecture notes:

$\lim_{x \to \infty} \sqrt{\sin{\frac{3}{\sqrt{x}}}} = \sqrt{\sin{3 \sqrt{ \lim_{x \to \infty} \frac{1}{x} }}}$

When can I move the $\lim$ into a function like this?

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    I'd prefer to write $\sin\left(3\sqrt{\bullet}\right)$ rather than $\sin 3\sqrt{\bullet}$, to make sure it wouldn't be mistaken for $\left(\sin 3\right)\sqrt{\bullet}$.2012-02-21

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Yes. Here is one (non-rigor) method of looking at it.

Let $\frac{1}{ x} = t$

As ${x \to \infty}, t \to 0 $

$\lim_{x \to \infty} \sqrt{\sin{\frac{3}{\sqrt{x}}}} = \lim_{t \to 0} \sqrt{\sin (3\sqrt{t})} = \sqrt{\sin(3 \times \lim_{t \to 0} \sqrt{t})}$ (Owing to Continuity)