Wednesday, 26 July 2017

Limit of limxto0+fracsinxsinsqrtx



How do I calculate this? lim
If I tried using l'Hopital's rule, it would become

\lim_{x\to0^+}\frac{\cos x}{\frac{1}{2\sqrt{x}}\cos \sqrt{x}}
which looks the same. I can't seem to find a way to proceed from here. Maybe it has something to do with \frac{\sin x}{x} \to 1
but I'm not sure what to do with it. Any advice?



Oh and I don't understand series expansions like Taylor's series.


Answer



By equvilency near zero \sin x\approx x we have
\lim_{x\to0^+}\frac{\sin x}{\sin \sqrt{x}}=\lim_{x\to0^+}\frac{x}{\sqrt{x}}=0
or
\lim_{x\to0^+}\frac{\sin x}{\sin \sqrt{x}}=\lim_{x\to0^+}\frac{\sin x}{x}\frac{\sqrt{x}}{\sin \sqrt{x}}.\sqrt{x}=1\times1\times0=0



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