Final answer:
L'Hôpital's Rule can be applied if we have an undefined numerator, but it depends on the denominator. The rule requires an indeterminate form like 0/0 or ∞/∞, and the existence of the limits of the derivatives of both the numerator and denominator.
Step-by-step explanation:
Can we apply L'Hôpital's Rule if we have an undefined numerator? The appropriate response to this question is: (c) It depends on the denominator. L'Hôpital's Rule is a powerful tool in calculus, particularly for evaluating limits that are in an indeterminate form such as 0/0 or ∞/∞. It states that if the limits of the functions in the numerator and the denominator both approach 0 or both approach infinity, then the limit of the quotient of the functions as x approaches a certain value can be found by taking the limits of the derivatives of the numerator and the denominator.
To apply L'Hôpital's Rule properly, you need an indeterminate form. If the numerator is undefined but the denominator is approaching a non-zero finite limit or infinity, L'Hôpital's Rule may not be applicable. However, if both the numerator and the denominator approach zero or both approach infinity, you can attempt to use L'Hôpital's Rule by differentiating the numerator and the denominator separately and then taking the limit.
It's also crucial to note that the rule is not a blanket solution: the existence of the limit of the derivatives is a precondition for the application of L'Hôpital's Rule. Moreover, even if the rule is applicable, one must ensure that repeated application of the rule leads to a limit that exists. Remember, while L'Hôpital's Rule can simplify the process of finding limits, it's not always the right or the only approach to evaluating a limit.