As x approaches 0 from the right, what is the limit of ln x?

As x approaches 0 from the right, what is the limit of ln x?

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

Answer: (A)

As x gets arbitrarily close to zero from the right, ln x decreases without bound. Since ln is defined only for positive x and ln x is negative for x < 1, shrinking x toward 0 pushes the natural logarithm to more and more negative values. For example, ln(0.1) ≈ -2.302, ln(0.01) ≈ -4.605, and this trend continues without limit. A handy viewpoint is x = e^{-t} with t → ∞, giving ln x = ln(e^{-t}) = -t → -∞. So the limit is negative infinity.

As x gets arbitrarily close to zero from the right, ln x decreases without bound. Since ln is defined only for positive x and ln x is negative for x < 1, shrinking x toward 0 pushes the natural logarithm to more and more negative values. For example, ln(0.1) ≈ -2.302, ln(0.01) ≈ -4.605, and this trend continues without limit. A handy viewpoint is x = e^{-t} with t → ∞, giving ln x = ln(e^{-t}) = -t → -∞. So the limit is negative infinity.