Evaluate the limit as x approaches 0 of (e^x - 1)/x.

Evaluate the limit as x approaches 0 of (e^x - 1)/x.

• A. 0
• B. -1
• C. 2
• D. 1

Answer: (D)

This limit tests the instantaneous rate of change of the exponential function at zero. It’s the derivative of e^x evaluated at x = 0. Since the derivative of e^x is e^x, at x = 0 this slope is e^0 = 1, so the limit is 1. Another way to see it is with the series expansion e^x = 1 + x + x^2/2 + ..., giving (e^x - 1)/x = 1 + x/2 + ..., which tends to 1 as x approaches 0. The limit reflects the slope of e^x at the origin, so it cannot be 0, -1, or 2. It must be 1.

This limit tests the instantaneous rate of change of the exponential function at zero. It’s the derivative of e^x evaluated at x = 0. Since the derivative of e^x is e^x, at x = 0 this slope is e^0 = 1, so the limit is 1. Another way to see it is with the series expansion e^x = 1 + x + x^2/2 + ..., giving (e^x - 1)/x = 1 + x/2 + ..., which tends to 1 as x approaches 0. The limit reflects the slope of e^x at the origin, so it cannot be 0, -1, or 2. It must be 1.