Key Idea: L'Hopital's rule finds limits that give 0/0 or ∞/∞. You show it by hand on Paper 1; the GDC's numerical limit does not count as using the rule.
L'Hopital's rule: the 0/0 form
- Substitute — Show that you get 0/0 or ∞/∞.
- Differentiate — Top and bottom separately, never the quotient rule.
- Substitute again — Read off the limit.
Repeated L'Hopital & rewriting forms
- Still 0/0? Differentiate again. Stop when you get a number.
- 0·∞: write f·g as f/(1/g) first.
- 'The limit exists': force the top to → 0 to find an unknown.
Derivatives that come up
- d/dx tan x = sec²x
- d/dx arctan x = 1/(1 + x²)
- d/dx sec x = sec x tan x
Check for 0/0 before every use of the rule.
Limit of (arctan 2x)/(tan 3x) as x → 0? 2/3.
(arctan(cos x) − k)/x² has a finite limit. Find k. k = arctan 1 = π/4.
And the limit itself? −1/4.
What is the classic zero-mark slip? Using the quotient rule instead of two separate derivatives.