This question is concerning a recent post: Find the follwing limit if $f$ is differentiable
Before looking at any of the answers posted I solved this problem using L'Hospital's rule, which results in the correct answer, as was noted by two other users. However, I'm unsure why the method is unsound. The individuals who used L'Hospital's rule in their solution had their answer downvoted. Specifically, @Paramanand Singh stated "L'Hospital's Rule does not apply here because we don't know if f(x) is differentiable at points other than x=0."
I went back to the textbook I was taught Calculus from, and it states L'Hospital's rule to be applicable if the functions f(x) and g(x) (making up the numerator and denominator of f(x)/g(x)) are differentiable on the interval (a,b), along with the necessary conditions mentioned in this post: When to Use L'Hôpital's Rule
My question, in response to @Paramanand Singh's statement: because we are given that f(x) is differentiable at x=0 (in the first link), would it be valid if I assumed f(x) is differentiable on the interval (0-ε,0+ε) and then applied L'Hospital's rule?
Mainly, could someone give me a more detailed explanation as to why L'Hospital's rule should not have been used?