Part 2/4:
Application of L'Hôpital's Rule
According to L'Hôpital's Rule, if we have an indeterminate form of the type ( 0/0 ), we can differentiate the numerator and the denominator until we eliminate the indeterminate form:
We first need to confirm that both functions in the numerator and the denominator indeed approach zero as ( x ) approaches zero. Graphical analysis confirms this fact – both branches approach zero from either side.
We then proceed by applying L'Hôpital's Rule. This means we will take the derivatives of both the numerator and the denominator as follows.
First Applications of L'Hôpital's Rule
Upon differentiating, we find that the first six derivatives of both the numerator and denominator still yield the limit of the indeterminate form ( 0/0 ).