What is lim_{x->∞} ln x?

What is lim_{x->∞} ln x?

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

Answer: (B)

As x grows without bound, the natural logarithm ln x grows without bound as well. In other words, ln x increases toward infinity. A handy way to see this is that for any large number M, you can choose x = e^M, and then ln x = M. This shows ln x can be made larger than any target, so the limit is infinity (in the extended real sense). The other possibilities don’t fit because the function does not settle at a finite value, nor does it head toward negative infinity. It also doesn’t fail to have a limit in the extended sense—it simply increases without bound.

As x grows without bound, the natural logarithm ln x grows without bound as well. In other words, ln x increases toward infinity. A handy way to see this is that for any large number M, you can choose x = e^M, and then ln x = M. This shows ln x can be made larger than any target, so the limit is infinity (in the extended real sense).

The other possibilities don’t fit because the function does not settle at a finite value, nor does it head toward negative infinity. It also doesn’t fail to have a limit in the extended sense—it simply increases without bound.