I've done problems exactly like these, but these two keep giving me the wrong answer, and I'm not sure why.
1) if |x - 2| < delta then |1/x - 0.5| < 0.2
2) if |x - 4| < delta then |sqrt(x) - 2| < 0.4
In both I need to find delta.
I've done problems exactly like these, but these two keep giving me the wrong answer, and I'm not sure why.
1) if |x - 2| < delta then |1/x - 0.5| < 0.2
2) if |x - 4| < delta then |sqrt(x) - 2| < 0.4
In both I need to find delta.
For 2.
Try multiplying $\displaystyle |\sqrt{x} - 2|$ by its conjugate.
Notice that
$\displaystyle |\sqrt{x} -2| < 0.4$
$\displaystyle |\sqrt{x} - 2| + 4 < 0.4 + 4$
$\displaystyle |\sqrt{x} + 2| < 4.4$.
So
$\displaystyle |\sqrt{x} - 2||\sqrt{x} + 2| < 0.4\cdot 4.4$
$\displaystyle |(\sqrt{x} - 2)(\sqrt{x} + 2)| < 1.76$
$\displaystyle |x - 4| < 1.76$.
So $\displaystyle \delta = 1.76$.