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. 1/2
• C. 1
• D. 2

Answer: (C)

This limit captures the slope of e^x at 0. It’s the derivative at 0, since the derivative is defined as lim as x approaches 0 of (f(x) − f(0)) / (x − 0). For f(x) = e^x, we have f(0) = 1 and f′(x) = e^x, so f′(0) = e^0 = 1. Therefore the limit is 1. You can also see this from the Taylor expansion e^x = 1 + x + x^2/2 + ..., which gives (e^x − 1)/x = 1 + x/2 + x^2/6 + ..., tending to 1 as x → 0.

This limit captures the slope of e^x at 0. It’s the derivative at 0, since the derivative is defined as lim as x approaches 0 of (f(x) − f(0)) / (x − 0). For f(x) = e^x, we have f(0) = 1 and f′(x) = e^x, so f′(0) = e^0 = 1. Therefore the limit is 1. You can also see this from the Taylor expansion e^x = 1 + x + x^2/2 + ..., which gives (e^x − 1)/x = 1 + x/2 + x^2/6 + ..., tending to 1 as x → 0.