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

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Prepare for the DAY 2002A Limits Test with our targeted quiz. Test your understanding with flashcards and multiple-choice questions. Each question features hints and explanations to enhance your learning. Ace your exam!

Multiple Choice

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

Explanation:
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.

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