MHB Double-Check My Problem Solving | Correct Reasoning Verification

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The discussion centers on verifying the correctness of a problem-solving approach regarding a continuous function at x=0. The initial poster acknowledges a misunderstanding of the question, clarifying that while the function is continuous, demonstrating that the limit as x approaches 0 equals 1 is necessary. Albert's reasoning is challenged, specifically regarding the inequalities he presented, which are incorrect near x=0. Additionally, there are errors in the limits he calculated, as they do not equal 1 but should be 0 instead. The conversation emphasizes the importance of accurately applying mathematical principles in problem-solving.
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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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