## Study23 3 years ago 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?

1. Study23

$$\ \Huge \left| \frac{1}{x} - \frac{1}{2} \right| < 0.2$$

2. satellite73

what are you trying to solve?

3. satellite73

i think you must start with $|\frac{2-x}{2x}|<0.2$

4. Study23

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 $$\ \delta$$.

5. satellite73

$-0.2<\frac{2-x}{2x}<0.2$ is the next step

6. satellite73

you probably are taking the limit as $$x\to 2$$ so you have control over the size of the numerator

7. Study23

Yes, I am taking the lim as x approaches 2

8. satellite73

or you can write $|\frac{2-x}{2x}|=\frac{1}{2}|\frac{x-2}{x}|<0.2$ so that $|\frac{x-2}{x}|<0.4$

9. Study23

If I do that, then where would I go from there?

10. satellite73

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 $|\frac{2-x}{x}|<3|x-2|$

11. Study23

How would I find $$\ \delta ?$$