Confusion over a limit. Different ways of solving give different answers? Qn: If it is given that
$$
\lim_{x\to\infty} \frac{x^2 - x - 2}{x + 1} - ax - b = 1
$$
then a and b must be?
Now, I tried doing this by 2 methods.
Method 1:
$$ \frac{x^2 - x - 2}{x + 1} - ax - b $$
$$ = (x - 2) - ax - b $$
Since the limit is finite, $a$ must be $= 1$ and so, $b = -3$
Method 2:
$$ \frac{x^2 - x - 2}{x + 1} - ax - b $$
$$ = \frac{x^2(1 - \frac 1 x - \frac2 {x^2})}{x(1 + \frac1x)} - ax - b $$
$$ = \frac{x - 1 - \frac2x}{(1 + \frac1x)} - ax - b $$
as $x \to \infty$, we have the above expression
$$ = x - ax - b$$
So, $a = 1$ and $b = -1$
Which of the above is correct?

Method $1$ is correct
In Method $2$, you cannot apply limit to a part of the fraction. So
$$ = \frac{x^2(1 - \frac 1 x - \frac2 {x^2})}{x(1 + \frac1x)} - ax - b $$
is not equal to 
$$ = x - ax - b$$
In the above step you have applied limit to just a part of that fraction leaving behind $\dfrac{x^2}{x}$.
hope the answer is clear !
