Find delta from a particular epsilon

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SUMMARY

The discussion focuses on finding delta (δ) from a given epsilon (ε) in the context of limits, specifically for the limit of x² as x approaches 3, which equals 9. The solution involves establishing the inequality |x² - 9| < 0.05, leading to the derived bounds of 2.9916 < x < 3.0083. The participant sought clarification on how the inequality 8.95 < x² < 9.05 translates to the bounds for x, which is resolved by taking the square root of each term. Ultimately, the calculated delta is δ = 0.0083.

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la_med12
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I'm learning how to find delta from a particular epsilon. I'm not understanding a step in the solution for the problem listed below:

Here's the problem:
lim(x→3)x^2=9

Solution:
|x^2-9|<.05
-.05<x^2-9<.05
2.9916..<x<3.0083
2.9916..-3<x-3<3.0083..-3
-.0083..<x-3<.0083..
delta=.0083

I don't understand how:
8.95<x^2<9.05
calculates to:
2.9916..<x<3.0083..
Could someone please explain this to me.. in simple terms. Cheers!
 
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Take the square root of each member.
 
Ah.. I guess I was looking at it for so long that I overlooked the obvious. Thanks for the help!
 

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