Compute lim_{x->∞} ln x.

Compute lim_{x->∞} ln x.

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

Answer: (B)

The main idea is that the natural logarithm increases without bound as its input grows. Since e^t grows without limit as t → ∞, its inverse function ln x must also increase without bound when x → ∞. More precisely, for any large number L, you can choose x = e^L, and then ln x = L; this shows ln x can exceed any target value, so the limit is infinity. The other possibilities don’t fit because ln x does not settle at a finite value like 0, it does not approach negative infinity since after x > 1 it becomes positive and keeps increasing, and the limit exists in the sense of diverging to infinity rather than failing to exist.

The main idea is that the natural logarithm increases without bound as its input grows. Since e^t grows without limit as t → ∞, its inverse function ln x must also increase without bound when x → ∞. More precisely, for any large number L, you can choose x = e^L, and then ln x = L; this shows ln x can exceed any target value, so the limit is infinity.

The other possibilities don’t fit because ln x does not settle at a finite value like 0, it does not approach negative infinity since after x > 1 it becomes positive and keeps increasing, and the limit exists in the sense of diverging to infinity rather than failing to exist.