L'Hôpital's rule is a limit tool for certain indeterminate forms, especially \(0/0\) and \(\infty/\infty\). When the hypotheses hold, the limit of \(f(x)/g(x)\) equals the limit of \(f'(x)/g'(x)\). Intuition: near the problem point, each function looks like its linear approximation, so the ratio of the functions behaves like the ratio of their slopes.

When you may use it

  • You are taking \(\lim_{x\to a} f(x)/g(x)\) (or \(x\to\pm\infty\)).
  • After substituting, you get \(0/0\) or \(\infty/\infty\) (other forms need rewriting first).
  • \(f\) and \(g\) are differentiable near \(a\) (except possibly at \(a\)), and \(g'(x)\neq 0\) in a punctured neighborhood.
  • The limit of \(f'/g'\) exists or is infinite — then the original limit matches it.

If the limit of \(f'/g'\) fails to exist as a real number or infinity, the rule does not conclude that the original limit fails; you need another method.

Quick sketch: \(0/0\)

Consider \(\lim_{x\to 0}\dfrac{\sin x}{x}\). Direct substitution is \(0/0\). Differentiating numerator and denominator gives \(\dfrac{\cos x}{1}\), and that limit as \(x\to 0\) is \(1\). So L'Hôpital recovers the standard result (you can also use the geometric squeeze proof; the rule is a convenient second path once derivatives are available).

Quick sketch: \(\infty/\infty\)

For \(\lim_{x\to\infty}\dfrac{\ln x}{x}\), substitution is \(\infty/\infty\). The derivative ratio is \(\dfrac{1/x}{1}=\dfrac{1}{x}\to 0\). Growth of \(x\) outpaces \(\ln x\).

Common traps

  • Applying the rule when the form is not indeterminate (for example \(1/0\) style blow-ups).
  • Differentiating the quotient with the quotient rule instead of separately differentiating numerator and denominator.
  • Stopping after one application when \(f'/g'\) is still indeterminate — you may need another round, or a rewrite (factor, conjugate, algebra).

Related: what a derivative measures, epsilon-delta intuition, and the calculus foundations topic.

Sources

Frequently asked questions

What is L'Hôpital's rule in one sentence?

When a limit of a quotient is an indeterminate form 0/0 or ∞/∞ and the hypotheses hold, the limit equals the limit of the quotient of the derivatives.

Does L'Hôpital's rule always work for 0/0?

Only when the stated differentiability and nonzero-denominator-derivative conditions hold and the limit of f'/g' exists (or is infinite). Otherwise use another method.

Should I use the quotient rule inside L'Hôpital's rule?

No. Differentiate the numerator and the denominator separately, then form the new quotient f'/g'.

What if f'/g' is still indeterminate?

You may apply the rule again if conditions still hold, or rewrite the expression algebraically before differentiating.