Strategies for Computing Limits Involving Square Roots and Rational Expressions

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Discussion Overview

The discussion revolves around strategies for computing limits involving square roots and rational expressions, specifically focusing on a limit that includes a summation and a square root term. Participants explore different approaches to evaluate the limit as \( n \) approaches infinity.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant seeks to isolate the variable \( i \) within the summation in the limit expression.
  • Another participant clarifies that \( i \) is a dummy variable for the summation and suggests that the limit can be approached via integration.
  • A participant expresses confusion about isolating \( i \) and questions how to handle the square root in the limit calculation.
  • One contributor asserts that isolating \( i \) is not feasible and suggests that a clever approximation might be necessary to compute the limit directly.
  • Another participant proposes that the limit likely relates to an integral, hinting at a specific integral that could simplify the evaluation process.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the feasibility of isolating \( i \) or the best method for computing the limit. Multiple approaches are discussed, indicating a lack of agreement on a singular method.

Contextual Notes

There are unresolved aspects regarding the assumptions needed for approximations and the specific methods for evaluating the limit, which depend on the interpretation of the summation and the square root term.

Who May Find This Useful

This discussion may be useful for students or individuals interested in advanced calculus, particularly those dealing with limits, summations, and integration techniques.

Caldus
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I'm trying to isolate the i variable in this formula. Thanks for any help.

[tex]\lim_{n->infinity}\frac{5}{n}\sum_{i=1}^{n}\sqrt{25-\frac{5i^2}{n^2}}[/tex]
 
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i is a dummy variable for the summation. What do you mean by "isolate"? Also by inspection, it looks like the limit can be gotten by integral from 0 to 1 of 5(25-5x2)1/2.
 
I'm trying to get i to be by itself inside the summation sign but I don't know how to get rid of that square root.
 
:confused:
I don't understand the last post. But if you want only to calculate the limit, mathman gave you the right direction (integrate)
 
You won't be able to isolate i in this equation.

To compute that limit directly, I think you'll pretty much have to do some sort of clever approximation, and prove the error in the approximation goes to zero as n goes to infinity. Actually carrying out this programme is well beyond what would be expected in a calc II Class.

This limit presumably comes from some sort of integral, probably [itex]\int_0^5 \sqrt{5^2 - x^2} \, dx[/itex]. There's a clever choice of partition for which the Riemann sum is much easier to compute, but as mathman was trying to hint, there's a much easier way to come up with the value of this integral...
 

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