Evaluate lim_{x→0} ln(1+x)/x.

Evaluate lim_{x→0} ln(1+x)/x.

• A. 1
• B. 0
• C. 1/2
• D. 2

Answer: (A)

This limit is about how fast ln(1+x) grows when x is very small. Although both the numerator and denominator go to zero, the change in the natural log near 1 is governed by its slope there. By the Mean Value Theorem, for some c between 0 and x we have ln(1+x) − ln(1) = (1/(1+c)) x. Since ln(1) = 0, this gives ln(1+x)/x = 1/(1+c). As x → 0, c → 0, so the expression approaches 1/(1+0) = 1. Equivalently, this is the derivative of ln at 1, which is 1. Therefore, the limit is 1.

This limit is about how fast ln(1+x) grows when x is very small. Although both the numerator and denominator go to zero, the change in the natural log near 1 is governed by its slope there. By the Mean Value Theorem, for some c between 0 and x we have ln(1+x) − ln(1) = (1/(1+c)) x. Since ln(1) = 0, this gives ln(1+x)/x = 1/(1+c). As x → 0, c → 0, so the expression approaches 1/(1+0) = 1. Equivalently, this is the derivative of ln at 1, which is 1. Therefore, the limit is 1.