Limit of a function in which square roots are involved $$\lim_{x \to 0}\frac{x+2-\sqrt{2x+4}}{3x-1+\sqrt{x+1}}$$
Could someone please help me solve this problem. 
I tried multiplying by a unity factor but I end up stuck. 

You may use $$\sqrt{1+x}=1+\frac x2+O(x^2)$$
First,
$$\sqrt{2x+4}=2\sqrt{1+\frac x2}=2(1+\frac x4)+O(x^2)=2+\frac x2+O(x^2)$$
And
$$\frac{x+2-\sqrt{2x+4}}{3x-1+\sqrt{x+1}}
=\frac{x+2-2-\frac x2+O(x^2)}{3x-1+1+\frac x2+O(x^2)}
\\=\frac{\frac x2+O(x^2)}{\frac72x+O(x^2)}=\frac17+O(x)\underset{x\to0}\longrightarrow\frac17
$$
