How to solve these limits?

Question

I can understand the concepts behind these limits, but I have no idea where to start to solve them.

Here are my questions

Answer 1

Start like this:

If $x\ge 0$ then $|f(x)-0|= |f(x)| = ... $

then check $x<0$...

$\frac17>|g(x)-\frac\pi2|=\frac\pi2-\arctan(2x) \ \iff \ \arctan(2x)>\frac\pi2-\frac17 \iff ...$

Answer 2

Look at specifically this part: $$|f(x)|<\frac{1}{100}.$$ If you are asked, "What is the largest value of $\delta$ such that $|x|<\delta$ implies (the above)," what is really being asked here? It's asking what the most basic inequality---as derived from the above---is. The $\delta$ is nearly there for confusion: It changes the question into a much more conceptually rich question than it actually is with the phrase, "The largest value of $\delta$ . . ."

The only tricky part, in my opinion, is observing that there are two definitions of $f(x)$ and noticing that there must, then, be two inequalities.

Likewise, focus on: $$\left|g(x)-\frac{\pi}{2}\right|<\frac{1}{7}.$$ What is the most basic inequality that can be derived from this that matches the format of $x>A$? The only thing to note here is that $A$ must be the smallest value possible.