calculus and analysis - How to evaluate the $0/0$ type limit in Mathematica?

When I use Limit to evaluate the limit

$$\begin{align}\lim_{k \to 0} \frac{ (k+2)(\alpha^2 - \sqrt{\alpha^4 + k}) + k}{\alpha^2 - \sqrt{\alpha^4 + k} + 2 k}\end{align}$$

((k + 2) (α^2 - Sqrt[α^4 + k]) + k)/(α^2 - Sqrt[α^4 + k] + 2 k)

(α and k are assumed to be real)

Limit gives the answer: 2.

However, I believe the correct answer is

(2 (-1 + α^2))/(-1 + 4 α^2)

So, when should I trust the answer of Limit?