But Why Does L'Hopital's Rule Work?

Why does the limit of the ratios equal the limit of the derivatives? Note: One viewer asked a great question in the comments: "For the ∞ / ∞ case, we showed the derivative outpaces the function in the provided example but why should it be true in general?" Answer: Here's an interesting way to think about it: we know that the function e^x has the property that its gradient is itself e^x. In other words, e^x exactly keeps pace with its own gradient. But e^x doesn't "grow fast enough" to have a vertical asymptote. So, in some sense, any function which grows fast enough to have a vertical asymptote must do so by growing an order of magnitude faster than its own value. Not a proof, I'll admit, but it helped me justify this idea in my own head. 0:00 Intro 2:00 The 0/0 Case 7:38 The ∞/∞ Case