Compute lim_{x->∞} (ln x) / x.

Compute lim_{x->∞} (ln x) / x.

• A. 1
• B. ∞
• C. Does not exist
• D. 0

Answer: (D)

A key idea here is that logarithmic growth is much slower than linear growth. As x becomes very large, ln x increases without bound, but x grows much faster, so their ratio ln x / x gets smaller and tends to zero. You can see this clearly with L’Hôpital’s rule. Since both the numerator and denominator go to infinity, differentiate top and bottom: the derivative of ln x is 1/x, and the derivative of x is 1. The limit becomes lim_{x→∞} (1/x) / 1 = lim_{x→∞} 1/x = 0. So the ratio vanishes in the long run. Intuitively, the slope of ln x at large x is 1/x, which shrinks to zero, while the denominator grows without bound, causing the overall fraction to approach zero.

A key idea here is that logarithmic growth is much slower than linear growth. As x becomes very large, ln x increases without bound, but x grows much faster, so their ratio ln x / x gets smaller and tends to zero.

You can see this clearly with L’Hôpital’s rule. Since both the numerator and denominator go to infinity, differentiate top and bottom: the derivative of ln x is 1/x, and the derivative of x is 1. The limit becomes lim_{x→∞} (1/x) / 1 = lim_{x→∞} 1/x = 0. So the ratio vanishes in the long run.

Intuitively, the slope of ln x at large x is 1/x, which shrinks to zero, while the denominator grows without bound, causing the overall fraction to approach zero.