Double-Check My Problem Solving | Correct Reasoning Verification

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Discussion Overview

The discussion revolves around verifying the reasoning behind solving a problem related to the continuity of a function and its limits, specifically at the point x=0. Participants are examining the conditions necessary to establish continuity and the implications of limits in this context.

Discussion Character

  • Debate/contested

Main Points Raised

  • One participant expresses uncertainty about their solution and seeks feedback on potential flaws in their reasoning.
  • Another participant asserts that the function is continuous and suggests substituting x = 0 as part of the verification process.
  • A later reply indicates a misunderstanding of the question by the initial poster and requests to disregard their previous comment.
  • Further, a participant clarifies that while the function is continuous at x=0, it is necessary to demonstrate that $\lim_{x\to 0} f(x) = 1$ to establish continuity, indicating that continuity alone is insufficient for this exercise.
  • Another participant challenges a claim regarding the inequalities involving x^2, pointing out that the stated inequalities do not hold true when x is close to 0, and questions the correctness of limits presented in earlier posts.

Areas of Agreement / Disagreement

Participants do not appear to reach a consensus, as there are competing views on the continuity of the function and the validity of the limits discussed. The discussion remains unresolved with differing interpretations of the problem.

Contextual Notes

There are limitations in the assumptions made regarding the function's behavior near x=0, and the discussion highlights potential misunderstandings about the application of continuity and limits.

Albert Einstein1
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I am not entirely sure if I solved this problem correctly. Please let me know if my reasoning is flawed. Thank you and I appreciate your help greatly.
 

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The function is continuous, just substitute x = 0...
 
On second thought, I didn't read the question properly... Please disregard my previous post.
 
The function IS continuous at x=0 but you need to show that $\lim_{x\to 0} f(x)= 1$ to show THAT so continuity cannot be used to do this exercise.

Albert, you state that $-1\le x^2- 2\le 1$. That is the same as $1\le x^2\le 3$. If x is close to 0 that is NOT true! You also have $\lim_{x\to 0} -x^2= 1$ and $\lim_{x\to 0} x^2= 1$. Those are certainly not true! Did you mean "= 0"?
 

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