"Let be continous and let be differenciable on (possibly at also, but we do not know this). Suppose also that exists. Prove that is differenciable at zero".
We want to show " is differenciable at zero", which means we want to show that the limit exists.
Note the numerator is . This is where we are using the countinuity condition. The denominator is zero as well. Thus, we can use L'Hopital here because the numerator is differenciable. Thus, . And this exists by the stated problem.