Finding the limit of a function . How can I calculate the following limit:
\begin{equation*}
\lim_{x \rightarrow a}
\frac{\sqrt{x} - \sqrt{a} + \sqrt{x-a} }{\sqrt{x^2 - a^2}}
\end{equation*}
I feel that I should multiply by the conjugate, but which conjugate? 

May be, you could make life a bit simpler using $x=y+a$ which makes
$$\frac{\sqrt{x} - \sqrt{a} + \sqrt{x-a} }{\sqrt{x^2 - a^2}}=\frac{\sqrt{a+y}-\sqrt{a}+\sqrt{y}}{\sqrt{y (2 a+y)}}$$ Now, use the generalized binomial theorem or Taylor series around $y=0$ to get 
$$\sqrt{a+y}=\sqrt{a}+\frac{y}{2 \sqrt{a}}+O\left(y^2\right)$$
$${\sqrt{y (2 a+y)}}= \sqrt{2a} \sqrt{y}+O\left(y^{3/2}\right)$$ which make finally 
$$\frac{\sqrt{a+y}-\sqrt{a}+\sqrt{y}}{\sqrt{y (2 a+y)}}=\frac{1}{\sqrt{2a} }+\frac{\sqrt{y}}{2 \sqrt{2}\,a}+O\left(y^1\right)$$ which shows the limit and also how it is approached when $y\to 0$.
