Calculate $\lim_{n\to\infty} (n - \sqrt {{n^2} - n} )$ Calculate limit: 
$$\lim_{n\to\infty} (n - \sqrt {{n^2} - n})$$
My try:
$$\lim_{n\to\infty} (n - \sqrt {{n^2} - n} ) = \lim_{n\to\infty} \left(n - \sqrt {{n^2}(1 - \frac{1}{n}} )\right) = \lim_{n\to\infty} \left(n - n\sqrt {(1 - \frac{1}{n}})\right)$$
$$\sqrt {(1 - \frac{1}{n}} ) \to 1$$
$$\lim_{n\to\infty} \left(n - n\sqrt {(1 - \frac{1}{n}})\right) = \lim_{n\to\infty} (n - n) = 0$$
Ok, it's not correct. In articles i found method:
$$\lim_{n\to\infty} \frac{{(n - \sqrt {{n^2} - n} )(n + \sqrt {{n^2} - n} )}}{{n + \sqrt {{n^2} - n} }} = \lim_{n\to\infty} \frac{n}{{n + \sqrt {{n^2} - n} }} = \lim_{n\to\infty} \frac{1}{{1 + \sqrt {1 - \frac{1}{n}}}} = \frac{1}{2}$$
Why the second method is valid, but the first one not? Whats difference?

The reason is that inside a limit, you can't substitute an expression with the limit it is approaching. The expresion $\sqrt{1-1/n}$ does go to $1$, but it does so at a slow pace, while the expression $n$ goes to infinity at a certain pace as well, so $n\sqrt{1-1/n}$ does not necessarily grow arbitrarily close to $n$.
