AHL 5.13 (HL)—L'Hopital's rule and limits

L’Hôpital’s rule compares leading limiting behaviour.

For suitable 0/0 or ∞/∞ forms, the ratio limit can equal the ratio of derivatives; the indeterminate form must be identified first.

Example

limₓ→0 sin x/x=lim cos x/1=1.

Verify the form and conditions before differentiating, then simplify and re-check the limit.

The rule is not a general quotient shortcut and does not apply directly to every finite ratio.

At a finite point or at infinity, first substitute and confirm an indeterminate $0/0$ or $\infty/\infty$ form. Then $\lim f/g=\lim f'/g'$ when the rule's conditions and the derivative-ratio limit hold; if the new ratio remains indeterminate, repeat. Alternatively use leading Maclaurin terms, for example $\sin x=x-x^3/6+\cdots$, so $\sin x/x\to1$. Products, differences and powers must first be algebraically converted to an eligible quotient or series form.