The limit lim_{x→0} ln(1+x) = 0 is justified by which property of the natural logarithm?

The limit lim_{x→0} ln(1+x) = 0 is justified by which property of the natural logarithm?

• A. Ln is continuous at 1 and ln(1)=0.
• B. Ln is differentiable at 0.
• C. Ln(1+x) tends to 1.
• D. Ln is defined for all real numbers.

Answer: (A)

The key idea is continuity of the natural logarithm at the input value 1. If you set t = 1 + x, then as x approaches 0, t approaches 1. Since ln is continuous at 1 and ln(1) = 0, the limit of ln(t) as t → 1 is ln(1) = 0. Hence ln(1+x) → 0. Differentiability would also imply continuity, but the direct justification here is the function’s continuity at 1. The claim that the limit is 1 is false, and ln is not defined for all real numbers (only positive inputs), so those options don’t justify the limit.

The key idea is continuity of the natural logarithm at the input value 1. If you set t = 1 + x, then as x approaches 0, t approaches 1. Since ln is continuous at 1 and ln(1) = 0, the limit of ln(t) as t → 1 is ln(1) = 0. Hence ln(1+x) → 0. Differentiability would also imply continuity, but the direct justification here is the function’s continuity at 1. The claim that the limit is 1 is false, and ln is not defined for all real numbers (only positive inputs), so those options don’t justify the limit.