Help with High-Power Limit Problem

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Discussion Overview

The discussion revolves around evaluating the limit as \( x \) approaches infinity for the expression \(\lim_{x\to \infty }\frac{(2x-1)^{10}(x-2)^{30}}{(x+1)^{20}(2x+3)^{20}}\). Participants explore various approaches and ideas related to this limit, focusing on techniques for simplifying the expression and identifying dominant terms.

Discussion Character

  • Homework-related
  • Mathematical reasoning
  • Exploratory

Main Points Raised

  • One participant expresses uncertainty about how to manipulate the expression, suggesting that the highest power, which is 30, might play a crucial role in the limit evaluation.
  • Another participant proposes a method of dividing each term in the expression by \( x \) to simplify the limit, rewriting the limit in terms of \( x^{-1} \) to analyze the behavior as \( x \) approaches infinity.

Areas of Agreement / Disagreement

Participants have not reached a consensus on the best approach to solve the limit. Multiple ideas and methods are presented, indicating ongoing exploration and discussion.

Contextual Notes

Some participants' suggestions depend on the assumption that the highest power in the numerator and denominator will dominate the limit, but this has not been conclusively established. The discussion includes various manipulations of the expression that may or may not lead to a definitive evaluation of the limit.

Yankel
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hello,

I need some help with this limit...

\lim_{x\to \infty }\frac{(2x-1)^{10}(x-2)^{30}}{(x+1)^{20}(2x+3)^{20}}

thanks...
 
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What ideas have you had so far?
 
The only thing I could think of was to somehow push x^30 inside the powers, but I don't know how...

30 is the strongest power when it comes to infinity, so there must be something with 30.
 
Yankel said:
hello,

I need some help with this limit...

\lim_{x\to \infty }\frac{(2x-1)^{10}(x-2)^{30}}{(x+1)^{20}(2x+3)^{20}}

thanks...
for \( x\ne 0 \) divide each of the 40 brackets at top and bottom by \(x\):

\[ \frac{(2x-1)^{10}(x-2)^{30}}{(x+1)^{20}(2x+3)^{20}}= \frac{(2-x^{-1})^{10}(1-2x^{-1})^{30}}{(1+x^{-1})^{20}(2+3x^{-1})^{20}}\]

CB
 

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