Ohio Assessments for Educators (OAE) Mathematics Practice Exam

What is the principle used to resolve limits that result in the form 0/0 or ∞/∞?

1. Factorization

2. Direct substitution

3. L'Hôpital's Rule

4. Graphical analysis

The correct answer is: L'Hôpital's Rule

When dealing with limits that yield indeterminate forms such as 0/0 or ∞/∞, L'Hôpital's Rule is the appropriate principle to apply. This rule states that if you have a limit that results in one of these indeterminate forms, you can differentiate the numerator and the denominator separately and then take the limit of the resulting expression. The utility of L'Hôpital's Rule lies in its ability to simplify the evaluation of limits that would otherwise be challenging to resolve directly. By applying the derivatives, it often transforms the limit into a more manageable form, possibly leading to a determinate value. In contrast, approaches like factorization might be effective in resolving certain algebraic limits but are not universally applicable and might not be practical or straightforward in all cases of indeterminate forms. Direct substitution can provide answers in many situations, but for the specific forms 0/0 or ∞/∞, it usually fails to yield a definitive result. Graphical analysis can offer insight into the behavior of functions but does not provide a systematic method for calculating limits directly, particularly in indeterminate forms. Thus, L'Hôpital's Rule is the most precise and effective method for resolving the specific indeterminate forms identified in the question.