as . So you can use L'Hospital's Rule here...
i) Let be differentiable, let . Suppose that exists.
Prove that .
ii) Suppose further that exists for all x and that exists. Prove that .
I realise I need to use L'Hopitals rule at some point but i'm not sure how I can justify the conditions needed to use it. If someone could help me with the first part, I may be able to get the second. Thanks!