Mathematics

Problem

Let

f(x)=\ln x-(x-1)

and

g(x)=x^{2}-1

. Consider the limit

\lim_{x\to 1}\frac{f(x)}{g(x)}.

Determine whether the conditions needed to replace this limit with the limit of the ratio

f'(x)/g'(x)

x=1

are satisfied. Respond with “yes” or “no,” and justify your answer by stating the relevant function values and derivative value(s) at

x=1