Question

Consider

Answer 1

The differential equation \[dy/dx = 1 - y\] be the particular solution to this differential equation with initial condition f(1) = 0. For thsi particularsolution f(x)>1 for all values of x.
Find \[\lim_{x \rightarrow 1} f(x)/(x^{3}-1)\]

Answer 2

to evaluate limit, apply L'hopitals rule
differentiate

--> f'(x)/3x^2 = 1-y/3x^2 = 1/3

Answer 3

It's f(x), not f'(x). Don't we only apply l'hopital's rule only when we know that f(x) is infinity when going infinity. Cases for infinity over infinity and 0 over 0... etc. ?

Answer 4

yes, but when you first evaluate the limit you get 0/0

Answer 5

But we don't know f(x). I got two functions after I integrated and solved for y.

Answer 6

This is the trouble I got in.

Answer 7

given f(1) = 0
thats all you need to know since the limit is as x->1

Answer 8

Ah! I didn't see that!

Answer 9

I'm so blind.

Answer 10

no i didn't see it at first either :)