Compute lim x->∞ (ln x)/x.

Compute lim x->∞ (ln x)/x.

• A. 0
• B. 1
• C. ∞
• D. -∞

Answer: (A)

Growth rates matter: as x grows, ln x increases without bound but much more slowly than x, so the ratio ln x / x shrinks toward zero. Formally, the limit is of the form ∞/∞; applying L’Hôpital’s rule gives the derivative of the numerator 1/x over the derivative of the denominator 1, yielding lim x→∞ (1/x) = 0. So the limit is 0.

Growth rates matter: as x grows, ln x increases without bound but much more slowly than x, so the ratio ln x / x shrinks toward zero. Formally, the limit is of the form ∞/∞; applying L’Hôpital’s rule gives the derivative of the numerator 1/x over the derivative of the denominator 1, yielding lim x→∞ (1/x) = 0. So the limit is 0.