Cue Column:

• When does L'Hopital apply?
• Common mistakes?
• When NOT to use it?

Notes Column: L'Hopital's Rule states: If lim(x->a) f$\frac{x}{g}$(x) gives 0/0 or infinity/infinity, then lim(x->a) f$\frac{x}{g}$(x) = lim(x->a) f'$\frac{x}{g}$'(x), provided the right-side limit exists (or is +/- infinity).

Conditions for validity: (1) Both f and g must be differentiable near a, (2) g'(x) != 0 near a (except possibly at a itself), (3) The limit of f'/g' must exist or be infinite.

Can be applied repeatedly if the result is still 0/0 or infinity/infinity. But each application requires re-checking the indeterminate form.

Common mistakes: (1) Applying to forms that are NOT indeterminate (e.g., 1/0 is not indeterminate, it's infinite). (2) Using the quotient rule instead of differentiating numerator and denominator separately. (3) Forgetting that L'Hopital gives no information if f'/g' limit doesn't exist.

When NOT to use: When algebraic simplification or standard limits give the answer faster. L'Hopital is a last resort, not the first tool.

Summary: L'Hopital works only for 0/0 and infinity/infinity. Differentiate numerator and denominator separately (NOT quotient rule). Verify indeterminate form before each application.