Evaluate the limit as x approaches infinity of (ln x) / x.

Evaluate the limit as x approaches infinity of (ln x) / x.

• A. 0
• B. 1
• C. Infinity
• D. -Infinity

Answer: (A)

The idea being tested is how different growth rates behave when put in a ratio. Linear growth in the denominator dominates the much slower growth of a natural logarithm in the numerator, so the fraction shrinks as x gets large. Because ln x increases without bound but far more slowly than x, their ratio must approach zero. A precise way to see this uses L’Hôpital’s rule: as x → ∞, both ln x and x go to infinity, so we can differentiate top and bottom. The derivative of ln x is 1/x and the derivative of x is 1, giving the limit lim x→∞ (1/x) = 0. Therefore the original limit is 0. It cannot be 1 or infinity, and since the expression is positive for large x, it cannot be negative infinity.

The idea being tested is how different growth rates behave when put in a ratio. Linear growth in the denominator dominates the much slower growth of a natural logarithm in the numerator, so the fraction shrinks as x gets large. Because ln x increases without bound but far more slowly than x, their ratio must approach zero.

A precise way to see this uses L’Hôpital’s rule: as x → ∞, both ln x and x go to infinity, so we can differentiate top and bottom. The derivative of ln x is 1/x and the derivative of x is 1, giving the limit lim x→∞ (1/x) = 0. Therefore the original limit is 0. It cannot be 1 or infinity, and since the expression is positive for large x, it cannot be negative infinity.