L'Hôpital's Rule for 0/0 Limits

This calculus question focuses on evaluating limits that at first glance are not written as quotients but can be turned into forms suitable for L'Hôpital's rule. The setups involve an exponential term compared against a polynomial term as the variable goes to infinity, and a logarithmic term interacting with a power of the variable near zero from the right. Candidates are expected to recognize hidden quotient structures, rewrite expressions appropriately, and decide whether L'Hôpital's rule is applicable from the behavior of numerator and denominator.

The techniques it leans on include algebraic rewriting to expose a quotient, careful handling of limits at infinity and at a one-sided limit, and repeated differentiation when the first application of L'Hôpital still leaves an indeterminate form. Interviewers watch for a clean justification of why L'Hôpital's rule applies, correct use of derivative rules for exponentials and logarithms, and a clear argument about which term dominates the growth or decay. Precision about domains, especially for the logarithm near zero, is also important.

What it tests

L'Hôpital's rule is a systematic method for evaluating limits that yield indeterminate forms like $0/0$ or $\infty/\infty$. The core insight is that, near the point of indeterminacy, the behavior of the ratio of two functions is governed by the behavior of their derivatives. This works because, locally, the linear (tangent) approximation to a function captures its growth or decay rate, so comparing derivatives tells us which function dominates as we approach the limit. The rule can be applied repeatedly if the indeterminate form persists after the first differentiation, as higher derivatives may reveal the dominant growth rate. The principle holds because differentiation preserves the essential rate-of-change information needed to resolve which function grows or decays faster.