The hint given was to take $\epsilon = 1$ for the formal definition of a limit. Doing this means that $|g(x) - M| < 1.$ Using the triangle inequality, you get $$|g(x)| = |(g(x) - M) + M| \le |g(x) - M| + |M| < 1 + |M|.$$ It also appears that the same method can be applied for $0 <\,\epsilon < 1.$
For $\epsilon = 0.5$, you end up getting $|g(x)| < 0.5 + |M| < 1 + |M|.$
How can the question be answered for $\epsilon > 1$?
You have already solved your question when you finish the use of triangle inequality. Why do you want to go for other values of $\epsilon$'s? The question is clearly not talking about any $\epsilon$. It is just asking you to find a $\delta > 0$ such that some specific statement containing $\delta$ is true. You have already shown that such a $\delta$ is there.
There is no need to think of anything further. Perhaps you have some misunderstanding of the definition of limit.
The definition of $\lim_{x \to a}f(x) = L$ says that for every $\epsilon > 0$ there exists a $\delta > 0$ such that the inequality $|f(x) - L| < \epsilon$ holds for all $x$ with $0 < |x - a| < \delta$.
The meaning of the above definition is somewhat difficult to grasp initially because the definition guarantees the truth of an infinite number of statements (one statement namely $|f(x) - L| < \epsilon$ for each $\epsilon > 0$). Also the truth of each such statement is not for all $x$ but for some specific values of $x$ based on $\epsilon$. This is where $\delta$ comes into picture. The definition thus says that for each $\epsilon$ the statement $|f(x) - L| < \epsilon$ is true provided we restrict $x$ to $0 < |x - a| < \delta$ for some specific number $\delta$ based on $\epsilon$. Thus the meaning of limit is very strong.
On the other hand your question gives you that $\lim_{x \to a}g(x) = M$. This means that each statement of type $|g(x) - M| < \epsilon$ is true for specific values of $x$ depending on $\epsilon$. Your question asks us to show the truth of just one of these statements. The student has to guess that the question is asking for the truth of the specific statement for $\epsilon = 1$. That's all the question is asking. It is not asking about the statements related to other values of $\epsilon$ (although by definition of limit they are also guaranteed to be true for specific values of $x$).