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The problem:

The Problem

As part of a homework assignment, we are required to fill in the blanks in these epsilon-delta proofs of limits.

I'm having trouble solving the inequality above, which is the primary problem. How do we deal with it? I have included the main question below in as well in case that changed anything.

Full Question:

Full Question

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  • $\begingroup$ 1-1/B is the answer I believe to be right, but I wanted to check it here before I submit $\endgroup$ – Delonjnaidu Mar 4 '17 at 15:01
  • $\begingroup$ You should please type your question in clear as part of the main text. $\endgroup$ – mlc Mar 4 '17 at 15:03
  • $\begingroup$ Right, sorry about that. It's my first time on here and I didn't want to make a mess of it so I thought including the pictures would make it easier on the eyes in case I did. $\endgroup$ – Delonjnaidu Mar 4 '17 at 15:04
  • $\begingroup$ is $$B>0$$ here? $\endgroup$ – Dr. Sonnhard Graubner Mar 4 '17 at 15:14
  • $\begingroup$ yes i have found it here $\endgroup$ – Dr. Sonnhard Graubner Mar 4 '17 at 15:14
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we have to solve $$\frac{1}{1-x^2}>B$$ with $$B>0$$ since we have $$B>0$$ then we have $$1-x^2>0$$ and we can multiply by $$1-x^2>0$$ and we get $$1>B(1-x^2)$$ dividing by $B$ we get $$\frac{1}{B}>1-x^2$$ from here we get $$x^2>1-\frac{1}{B}$$ if $$B\le 1$$ then the inequality is true, else if $$B>1$$ then we have $$|x|>\sqrt{1-\frac{1}{B}}$$ which is given in the Statement above

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