Today we explore more of L’Hôpital’s rule. The idea is that we can replace a function with its tangent line approximation and use this to evaluate the limit. But more precisely:
Let \(f\) and \(g\) be differentiable on an open interval that includes \(x = a\) and suppose that \(f(a) = g(a) = 0\) and that \(g'(a) \ne 0\text{.}\) Then
Another use case for L’Hôpital’s rule is to find “end behavior” of functions. That is, what happens to a function as \(x\) gets arbitrarily large. We write this as \(\lim_{x \to \infty}f(x)\text{.}\) Here are some examples.
Find \(\d\lim_{x\to \infty} \frac{\ln(x)}{2\sqrt{x}}\text{.}\) What does this say about that function, and about the numerator and denominator separately.
