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?
 A: 
$$\lim (n - n\sqrt {(1 - \frac{1}{n}} )) = \lim (n - n) = 0$$

This is not correct. If $n$ goes to infinity, all $n$'s go to infinity, not just the one in $\sqrt{1-\frac{1}{n}}$. Here it's as if you fixed the first two $n$'s of your expression, and let the third one go to infinity.
So with this method you would get $\infty - 1\times\infty$, which is indeterminate.
A: 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 $n$.
A: The problem is this step:
$$\lim (n - n\sqrt {(1 - \frac{1}{n}} )) = \lim (n - n)$$
How do you justify this? You need to take the limit of everything all at once.
A: Given $\displaystyle \lim_{n\rightarrow \infty}\left[n-\sqrt{n^2-n}\right] = \lim_{n\rightarrow \infty}\left[n-n\cdot \left(1-\frac{1}{n}\right)^{\frac{1}{2}}\right]$
Now Using $\displaystyle (1+x)^{n} = 1+nx+\frac{n(n-1)}{2}x^2+.....$
So we get $\displaystyle \lim_{n\rightarrow \infty}\left[n-n\left(1-\frac{n}{2n}+\frac{1}{2}\cdot \frac{1}{2}\cdot \frac{1}{2n^2}+.......\infty\right)\right] = \frac{1}{2}$
A: Note that your method is wrong:
$$
\lim_{n\rightarrow +\infty}{n(1-\sqrt{1-\frac1{n}}}=0\times \infty
$$
Which is undefined
