Evaluate lim x→∞ (ln x)^2 / x.

Evaluate lim x→∞ (ln x)^2 / x.

• A. ∞
• B. -∞
• C. 1
• D. 0

Answer: (D)

This question hinges on how slowly logarithmic growth happens compared to linear growth. As x becomes very large, ln x grows without bound but at a pace far, far slower than x, so even when you square the logarithm, it’s still tiny next to the linear x in the denominator. A solid way to see this is by L’Hôpital’s rule: start with lim (ln x)^2 / x. Differentiating top and bottom gives lim (2 ln x)/x. Apply L’Hôpital again: lim 2/x, which goes to 0 as x → ∞. So the original limit is 0.

This question hinges on how slowly logarithmic growth happens compared to linear growth. As x becomes very large, ln x grows without bound but at a pace far, far slower than x, so even when you square the logarithm, it’s still tiny next to the linear x in the denominator. A solid way to see this is by L’Hôpital’s rule: start with lim (ln x)^2 / x. Differentiating top and bottom gives lim (2 ln x)/x. Apply L’Hôpital again: lim 2/x, which goes to 0 as x → ∞. So the original limit is 0.