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.

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- Jun 9th 2009, 07:04 AMfardeen_genGreatest integer and modulus function limit?
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.

- Jun 10th 2009, 02:57 PMthe_doc
I don't believe this function's limit can be defined from the left as the greatest integer function jumps from when x is just less than 0 to when it is 0.

- Jun 11th 2009, 02:53 AMMoo
- Jun 11th 2009, 04:47 AMthe_doc
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

$

$\displaystyle \lim_{x \to 0^{-}} \frac{\sum_{r = 1}^{2n + 1} [x^r] + (n + 1)}{1 + [x] + |x| + 2x}= 0$