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?
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?
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)