Compute lim_{x->0^+} ln x.

Compute lim_{x->0^+} ln x.

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

Answer: (A)

As x approaches 0 from the right, the natural log of x heads toward negative infinity. The reason is that ln x is defined only for positive x, and numbers smaller than 1 have negative logarithms. As you push x smaller and smaller (0.1, 0.01, 0.001, …), the values of ln x become increasingly negative: -2.302, -4.605, -6.907, and so on without bound. Since you can make ln x as negative as you like by choosing x close enough to 0, the limit is negative infinity (in the extended real sense). The function isn’t defined for x ≤ 0, so the left-hand side isn’t relevant here, and the limit isn’t a finite number.

As x approaches 0 from the right, the natural log of x heads toward negative infinity. The reason is that ln x is defined only for positive x, and numbers smaller than 1 have negative logarithms. As you push x smaller and smaller (0.1, 0.01, 0.001, …), the values of ln x become increasingly negative: -2.302, -4.605, -6.907, and so on without bound. Since you can make ln x as negative as you like by choosing x close enough to 0, the limit is negative infinity (in the extended real sense). The function isn’t defined for x ≤ 0, so the left-hand side isn’t relevant here, and the limit isn’t a finite number.