Help solve a Limit Question?

Question

See this .

What he's meant that "in particular"? where the $|g(x)|<|M|+1$ formula from? How deduced? What is the meaning of it?

Answer 1

You have $$ M-1<g(x)<M+1 $$ Now use the fact that $-|M|\le M\le |M|$, so $$ -|M|-1\le M-1<g(x)<M+1\le |M|+1 $$ and the external terms tell you that $$ |g(x)|<|M|+1 $$

The whole point is to prove that $g$ is bounded in a neighborhood of $a$.

Answer 2

One form of the triangle inequality is $|a|-|b|\le|a-b|$. Thus

$$|g(x)-M|\lt1\implies|g(x)|-|M|\le|g(x)-M|\lt1\implies|g(x)|\lt|M|+1$$