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Question Number 186748 by EnterUsername last updated on 09/Feb/23

$$\mathrm{Let}f:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}\mathrm{be}\mathrm{a}\mathrm{function}\mathrm{satisfying}\mathrm{the}\mathrm{relation}$$$$f(x.f\left(\mathrm{y}\right))=f\left(x\mathrm{y}\right)+x\mathrm{for}\mathrm{all}x,\mathrm{y}\in {\mathbb{R}}^{+}.\mathrm{Then}$$$$\underset{x\to 0}{\lim }\left(\frac{{\left(f\left(x\right)\right)}^{1/3}-1}{{\left(f\left(x\right)\right)}^{1/2}-1}\right)=$$$$\left(\mathrm{A}\right)1\left(\mathrm{B}\right)\frac{1}{2}$$$$\left(\mathrm{C}\right)\frac{2}{3}\left(\mathrm{D}\right)\frac{3}{2}$$

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