What is lim_{x->∞} (ln x) / x?

What is lim_{x->∞} (ln x) / x?

• A. It converges to 0
• B. It diverges to ∞
• C. It diverges to -∞
• D. It oscillates

Answer: (A)

The main idea is how fast each part grows: x increases linearly while ln x grows much more slowly. As x gets large, the denominator outpaces the numerator, so the ratio shrinks toward zero. This can be made rigorous with L'Hôpital's rule since the expression is in an ∞/∞ form. Differentiating top and bottom gives (1/x)/1, and taking the limit as x→∞ yields 0. So the limit is 0.

The main idea is how fast each part grows: x increases linearly while ln x grows much more slowly. As x gets large, the denominator outpaces the numerator, so the ratio shrinks toward zero. This can be made rigorous with L'Hôpital's rule since the expression is in an ∞/∞ form. Differentiating top and bottom gives (1/x)/1, and taking the limit as x→∞ yields 0. So the limit is 0.