Full answers! need all the i can get

This critter is really a nasty piece of work. If you have a sister or daughter, I'd never let her date this question.

The question is What happens at 4 -- exactly.
The function at x =4 is determined by 2 * x or 2*4 = 8

But look what happens to the top and bottom functions when you get infinitely close to x = 4. You could let x = 3.99999 to see what happens to the top function

g(x) = x - 2 when x is less than 4. So ...
Put in g(3.99999) = x - 2
g(3.99999) = 3.99999 - 2 = 1.99999 which rounds to 2. That's nowheres near 8.

We have already shown that at this point this is going to be discontinuous. 2 does not equal 8. But just for fun, we should try the bottom function
Suppose x = 3.99999 again.
then g(3.99999) = sqrt(3.99999) which is almost 2. there's no way that will work.

So confused right here i need it to be more clearer