Why replacing two functions by their derivatives works, when you are allowed to do it, and when you are not.
Both of these curves pass through zero at the same place, so the ratio is 0⁄0 and tells you nothing. Zoom in and watch them both straighten into their tangents. The ratio of two straight lines through the same point is just the ratio of their gradients.
Zoom all the way in and the two readouts meet.
If lim f(x)g(x) gives 0⁄0 or ∞⁄∞, then it equals lim f′(x)g′(x), provided that second limit exists.
Check the form first, every time. The rule applies only to 0⁄0 and ∞⁄∞. Applying it to something like 3⁄2 gives a completely wrong answer, and writing "0/0, so by l'Hopital" is usually worth a mark on its own.
It is not the quotient rule. Differentiate the top and the bottom separately. The two things look similar on the page and are completely different operations.
Repeated use is allowed and expected, but check the form again before each application. The moment it stops being indeterminate, stop differentiating and substitute.
1. limx→0 ex − 1x is:
2. For limx→1 x + 2x + 1, what should you do?
3. How many times must you apply the rule to limx→0 1 − cos xx²?
Stating the indeterminate form before using the rule. One line, often one mark, and it is what shows you checked rather than guessed.
Differentiating top and bottom separately and correctly. The quotient rule here is a method error, not an arithmetic one.
Re-checking the form between repeated applications, and stopping as soon as substitution works.
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