Question

lim x->0^+ (sinx)/(x+sqrootx)

Answer 1

There's always L'hôpital for your everyday indeterminate needs...

Answer 2

I you don't know L'Hopital's rule yet: For \(x\) near 0, you have \(\sin x\approx x\).
\[\lim_{x\to0^+}\frac{\sin x}{x+\sqrt x}=\lim_{x\to0^+}\frac{x}{x+\sqrt x}=\lim_{x\to0^+}\frac{1}{1+x^{-1/2}}\]
Multiply numerator and denominator by \(\sqrt x\):
\[\lim_{x\to0^+}\frac{1}{1+x^{-1/2}}\cdot\frac{\sqrt x}{\sqrt x}=\lim_{x\to0^+}\frac{\sqrt x}{\sqrt x+1}\]