Find parameters in limit Find real a,b so that
$$
\lim_{n\to\infty} \sqrt{n^2-n+1}-an-b =-\infty
$$
 I have no idea what to do.
Thank you.
 A: You can rewrite this as
$$
n\left (\sqrt{1 - \frac{1}{n} + \frac{1}{n^2}} - a\right) - b
$$
As $n\to \infty$, the only term that really matters is the first one.

Hence, $b$ can be any real number.

Now the term under the square root  gets very close to 1, and if the limit has to be $-\infty$, then the term
$$
t_n := \sqrt{1 - \frac{1}{n} + \frac{1}{n^2}} - a
$$
must be negative at some point. Note that
$$
\lim_{n\to \infty} t_n = 1 - a
$$
Hence, if $1-a < 0$, then $t_n < 0$ for large $n$.

Hence, for the limit to be $-\infty$, it must happen that $a>1$

A: Putting $n=\frac1h,$
$$A=\lim_{n\to\infty}\sqrt{n^2-n+1}-an-b=\lim_{h\to0}\frac{(1+h^2-h)^{\frac12}-a-bh}h$$
(Using Generalized Binomial Expansion)
$$A=\lim_{h\to0}\frac{1+\frac{h^2-h}2+O(h^2)-a-bh}h$$ 
$$=\lim_{h\to0}\left(\frac{1-a}h+O(h)\right)-b-\frac12$$
So, we need $1-a<0$
A: Consider how the square root with everything inside it tends to approach $n$ as $n$ goes to infinity. Thus, $$\lim_{n\to\infty} n - an - b \Longrightarrow n(1 - a) - b = -\infty$$
for what values of $n$?
A: The parameter $b$ is irrelevant: if the limit is inexistent/finite/$\infty$/$-\infty$ without it it will be in the same class with it.
What happens when $a\le0$?
And when $a>0$, try considering
$$
\sqrt{n^2-n+1}-an=
\frac{(\sqrt{n^2-n+1}-an)(\sqrt{n^2-n+1}+an)}{\sqrt{n^2-n+1}+an}=
\frac{n^2-n+1-a^2n^2}{\sqrt{n^2-n+1}+an}
$$
A: Factor an $n$ out of the expression as follows:
$$n\left(\sqrt{1-\frac{1}{n}+\frac{1}{n^2}}-a-\frac{b}{n}\right)$$
Now you can see that the remaining square root approaches 1, and the $\frac{b}{n}$ term approaches $0$.  If $a>1$, then the expression in the parentheses is negative for large $n$, so the limit is $-\infty$.  If $a<1$ then the expression is positive for large $n$, so the limit is $+\infty$.  In both cases $b$ can be anything.
