I found a proof for the vector chain rule, but it makese no sense to me

[mr_coffee]

Hello everyone, our professor wanted us to find the vector chain rule proof and i found one here:
http://web.mit.edu/wwmath/vectorc/scalar/chain.html
But it makes no sense to me, where are the limits?

 

[JoAuSc]

Taking the limit is implied when the author says "Now as we let delta t go to zero". This kind of proof relies a bit more on mathematical intuition than the definition for the derivative you learn in Calc I. For one thing, it implies you're familiar with approximating things by Taylor series. Which part of the proof are you having trouble with?

 

[mr_coffee]

hah well i did take that in calc II but forgot even how that works, but i don't get how he starts off saying, let's first notice a few things... x(t+change in t) = x + chag in x
then he goes to say:
now by linear approximation...what the? is there another proof that goes through the limits, i understand that a lot better. He told us to find proofs and make sure we understand them, so he isn't grading this, but he will probably tell us to prove it on an exam