Answer
(a) The interval is $(0.16,0.36)$
(b) $\delta=0.09$
Work Step by Step
Find a $\delta\gt0$ such that for all $x$ $$0\lt |x-\frac{1}{4}|\lt\delta\Rightarrow|\sqrt{x}-\frac{1}{2}|\lt0.1$$
1) Find the interval around $0$ on which $|\sqrt{x}-\frac{1}{2}|\lt0.1$ holds.
Solve the inequality: $$|\sqrt{x}-\frac{1}{2}|\lt0.1$$ $$-0.1\lt\sqrt x-\frac{1}{2}\lt0.1$$ $$0.4\lt\sqrt x\lt0.6$$
Square: $$0.16\lt x\lt0.36$$
The open interval around $1/4$ is $(0.16,0.36)$.
2) Give a value for $\delta$
The nearer endpoint to $0$ is $0.16$, and the distance between them is $1/4-0.16=0.09$.
So if we take $\delta=0.09$ or any smaller positive number, then $0\lt|x-1/4|\lt0.09$, meaning all $x$ would be placed in the interval $(0.16,0.36)$ so that $|\sqrt{x}-\frac{1}{2}|\lt0.1$.
In other words, $$0\lt |x-\frac{1}{4}|\lt0.09\Rightarrow|\sqrt{x}-\frac{1}{2}|\lt0.1$$
