How close to 1 does $x$ have to be to ensure that the given function is within a distance of $0.5$ of its limit

Question

I hemmed and hawed over whether I should ask about this. The question seems clear enough, but it just seems far too easy. I'm worried I'm misunderstanding it.

I am confused about part b of this question in my textbook.

Part a

Use numerical and graphical evidence to guess the
value of the limit $$\underset{x \to 1} \lim \frac{x^3-1}{\sqrt{x}-1}$$

part b

How close to $1$ does $x$ have to be to ensure that the
fun­ction in part a is within a distance $0.5$ of its limit?

So basically, as I understand it, $x$ would have to be in the range $(0.5, 1.5)$ to be within a distance of $0.5$ of $1$.

I must be misunderstanding this, right?

Answer 1

We know the limit is $6$.

The question is asking us to determine $\delta>0$ such that $0<|x-1|<\delta$ would ensure that

$$\left|\frac{x^3-1}{\sqrt{x}-1} -6\right| < \frac12$$

This picture might help you.