How do I evaluate the limit? I am trying to evaluate limit$$\lim_{x\to 0}\frac{\sqrt{x+2}-\sqrt{2}}{x}$$ For my answer I got $\frac{1}{2\sqrt{2}}$. My professor answer sheet says $\frac{\sqrt2}{4}$. I am not sure why that is. My work and problem is attached below. Thank you.


Your answers are equivalent, as noted in the comments.
$$\frac{1}{2\sqrt 2} = \frac{1}{2 \sqrt 2} \cdot \underbrace{\frac{\sqrt 2}{\sqrt 2}}_{=1} = \frac{\sqrt 2}{2 \sqrt 2 \sqrt 2} = \frac{\sqrt 2}{2 \cdot 2} = \frac{\sqrt 2}{4}$$
