Proving a Limit: Epsilon & Delta Solution

Thread starter ace123
Start date Sep 15, 2007
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Limit

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SUMMARY

The discussion focuses on proving the limit of the function \(\lim_{x \rightarrow 5} \frac{x^3-6x^2+15x-47}{x^2-8x+14} = -3\) using the epsilon-delta definition. Participants clarify the process of bounding the numerator and denominator separately, emphasizing the importance of selecting appropriate intervals for \(x\) to ensure the limit is correctly proven. The final approach involves establishing that \(|f(x) + 3| < \epsilon\) by manipulating the inequalities derived from the limit definition.

PREREQUISITES

Epsilon-delta definition of limits
Factoring polynomials
Understanding of rational functions
Basic calculus concepts

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Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of limit proofs using the epsilon-delta method.

Sep 15, 2007

ace123

learningphysics said:

|(x^2+2x+1)/(x^2-8x+14)| is bounded above by 42.25/1.75 over the range of |x-5|<1/2. If we didn't know that |x-5|<1/2, we wouldn't be able to use this bound.

Thanks. you're welcome. I really appreciate the compliment. But don't take my word for anything, or anyone else... convince yourself... it's possible I made a mistake. anyway, good luck on your exam!

Oh I understand now. I guess i completely forgot that we had to restrict x to the 1/2 interval. It makes complete sense now. I really appreciate you help and thanks