As x approaches infinity, what is the limit of (1 + 1/x)^x?

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Multiple Choice

As x approaches infinity, what is the limit of (1 + 1/x)^x?

Explanation:
This limit is the classic way e is defined. Let y = (1 + 1/x)^x. Take natural logs: ln y = x ln(1 + 1/x). Let t = 1/x, so t → 0+. Then ln y = (1/t) ln(1 + t) = ln(1 + t)/t. We know that lim_{t→0} ln(1 + t)/t = 1, so ln y → 1. Exponentiating gives y → e^1 = e. Therefore, as x grows without bound, (1 + 1/x)^x approaches e.

This limit is the classic way e is defined. Let y = (1 + 1/x)^x. Take natural logs: ln y = x ln(1 + 1/x). Let t = 1/x, so t → 0+. Then ln y = (1/t) ln(1 + t) = ln(1 + t)/t. We know that lim_{t→0} ln(1 + t)/t = 1, so ln y → 1. Exponentiating gives y → e^1 = e. Therefore, as x grows without bound, (1 + 1/x)^x approaches e.

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