Answer
$\dfrac{x+\sqrt{x^2-h^2}}{h}$
Work Step by Step
Rationalize the denominator by multiplying $\sqrt{x+h}+\sqrt {x-h}$ to both the numerator and the denominator:
$\dfrac{\sqrt{x+h}+\sqrt {x-h}}{\sqrt{x+h}-\sqrt {x-h}}\\$
$=\dfrac{\sqrt{x+h}+\sqrt {x-h}}{\sqrt{x+h}-\sqrt {x-h}}\cdot \dfrac{\sqrt{x+h}+\sqrt {x-h}}{\sqrt{x+h}+\sqrt {x-h}}$
$=\dfrac{\left(\sqrt{x+h}+\sqrt {x-h}\right)^2}{\left(\sqrt{x+h}-\sqrt {x-h}\right)\left(\sqrt{x+h}+\sqrt {x-h}\right)}$
Use special formula $(a+b)^2=a^2+2ab+b^2$ in the numerator and $(a+b)(a-b)=a^2-b^2$ in the denominator.
$=\dfrac{\left(\sqrt{x+h}\right)^2+2(\sqrt{x+h})(\sqrt {x-h})+\left(\sqrt {x-h}\right)^2}{\left(\sqrt{x+h}\right)^2-\left(\sqrt {x-h}\right)^2}$
$=\dfrac{x+h+2(\sqrt{x+h})(\sqrt {x-h})+x-h}{x+h-(x-h)}$
$=\dfrac{2x+2(\sqrt{x+h})(\sqrt {x-h})}{x+h-x+h}$
$=\dfrac{2x+2(\sqrt{x+h})(\sqrt {x-h})}{2h}$
Use the rule $\sqrt{a\cdot b}=\sqrt{a} \cdot \sqrt{b}$ then simplify:
$=\dfrac{2x+2\sqrt{(x+h)(x-h)}}{2h}$
$=\dfrac{2x+2\sqrt{x^2-h^2}}{2h}$
Factor out $2$ in the numerator then simplify.
$=\dfrac{2(x+\sqrt{x^2-h^2})}{2h}$
$=\dfrac{x+\sqrt{x^2-h^2}}{h}$
Hence, the correct answer is $\frac{x+\sqrt{x^2-h^2}}{h}$.
