Evaluate:
$\displaystyle \lim_{x\rightarrow 0^{-}} \frac{\sum_{r = 1}^{2n + 1} [x^r] + (n + 1)}{1 + [x] + |x| + 2x}$
where $\displaystyle [.]$ denotest the greatest integer function.
Had a look at the question again and you're correct Moo - I misread. It does exist as we only need to get arbitrarily close to x=0 to define the limit.
In which case we can consider the range $\displaystyle -1 < x < 0$ where
$\displaystyle -1 < x^r < 0$ for $\displaystyle r$ odd $\displaystyle \Rightarrow [x^r ] = -1$
$\displaystyle 0 < x^r < 1$ for $\displaystyle r$ even $\displaystyle \Rightarrow [x^r ] = 0
$ so within this range
$\displaystyle \sum_{r=1}^{2n+1} [x^r ] = -(n+1) .$
Also in this range we have that
$\displaystyle 1 + [x] + |x| + 2x = 1 -1 -x + 2x = x$
and so since
$\displaystyle \frac{0}{x} = 0$ when $\displaystyle x<0
$
$\displaystyle \lim_{x \to 0^{-}} \frac{\sum_{r = 1}^{2n + 1} [x^r] + (n + 1)}{1 + [x] + |x| + 2x}= 0$