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Study23
How do I isolate the x in this expression? (ie. so x is by itself). In other words, how do I factor that fraction out?
\(\ \Huge \left| \frac{1}{x} - \frac{1}{2} \right| < 0.2 \)
what are you trying to solve?
i think you must start with \[|\frac{2-x}{2x}|<0.2\]
It's in the format of \(\ \Huge \left| f(x) - L \right| < \epsilon \). Im trying to factor that expression so I can use the end result to find \(\ \delta \).
\[-0.2<\frac{2-x}{2x}<0.2\] is the next step
you probably are taking the limit as \(x\to 2\) so you have control over the size of the numerator
Yes, I am taking the lim as x approaches 2
or you can write \[|\frac{2-x}{2x}|=\frac{1}{2}|\frac{x-2}{x}|<0.2\] so that \[|\frac{x-2}{x}|<0.4\]
If I do that, then where would I go from there?
you have control over \(|x-2|\) that is you can make it as small as you like as for the \(x\) in the denominator, you can simply say that since you are taking the limit as \(x\to \frac{1}{2}\) you can assert that it is say between \(\frac{1}{3}\) and \(\frac{2}{3}\) so that the whole thing will be largest when the denominator is smallest, namely when it is \(\frac{1}{3}\) giving the inequality \[|\frac{2-x}{x}|<3|x-2|\]
How would I find \(\ \delta ?\)