Question

Calculus: How do I know which theorems I am using?

Answer 1

You only show Theorem 6. I assume there are more?

Answer 2

Is there a theorem that says the square root or cube root function is continuous?

Answer 3

I am not sure...but it might be the composite theorems. Also, ignore the titles at the tops of the yellow boxes. (Theorem 6, Theorem 7, etc.) they are only there for organization. All of the theorems I have provided are for continuous functions.

Answer 4

For 31) I can imagine saying this:
f(x) = x is continuous (we need a reason)

f1(x) = x+2 is continuous : Let g(x)=2 f(x)+g(x) is continuous by Thm 6, part 1
f2(x) = sqr(x+2) is continuous (we need a reason, composite function ?)
f3(x) = 1/sqr(x+2) is continuous. Let g(x)=1. then g(x)/f2(x) is continuous Thm 6, 5 for x>-2


That is my best guess as to how to do this

Answer 5

That happens to be the answer. I think I'm understanding this now. :)