Conjugate error - What am I doing wrong?

Question

I have this equation and I'm multiplying a conjugate but I'm not getting the right answer.

$$f(x)=x\sqrt{x}$$ $$\lim_{h \to 0}{(x+h)(\sqrt{x+h})-x\sqrt{x}\over h} \times {\sqrt{x+h}+\sqrt{x}\over\sqrt{x+h}+\sqrt{x}}$$ $$\lim_{h \to 0}{x^2+2xh+h^2-x^2\over{h(\sqrt{x+h}+\sqrt{x}})}$$ $$\lim_{h \to 0}{h(2x+h)\over{h(\sqrt{x+h}+\sqrt{x}})}$$ $$2x\over2\sqrt{x}$$ instead of $${3\sqrt{x}\over2}$$

Answer 1

you must multiply numerator and denominator by $$(x+h)\sqrt{x+h}+x\sqrt{x}$$

Answer 2

For all $x>0$ it should be $$(\sqrt{x^3})'=\lim_{h\rightarrow0}\frac{\sqrt{(x+h)^3}-\sqrt{x^3}}{h}=\lim_{h\rightarrow0}\frac{(x+h)^2+(x+h)x+x^2}{\sqrt{(x+h)^3}+\sqrt{x^3}}=\frac{3x^2}{2\sqrt{x^3}}=\frac{3}{2}\sqrt{x}.$$

Answer 3

You need to fix here $$\lim_{h \to 0}{ (x+h) (\sqrt{x+h})-x\sqrt{x}\over h} \cdot {\color{red}{(x+h)} \sqrt{x+h}+\color{red}{x} \sqrt{x}\over\color{red}{(x+h)} \sqrt{x+h}+\color{red}{x} \sqrt{x}}$$