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Here is the problem: (1 plus 1/x) to the power x.

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Find the limit of (1 plus 1 / x) to the power x as x → ∞.

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Indeterminate form 1^∞ — take logarithms. Write the power as e to the power exponent times ln base); since eᵘ is continuous, the limit is e raised to the limit of the exponent.

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Indeterminate form 0 times ∞ — rewrite as a quotient. Moving one factor to the denominator as its reciprocal turns 0 times ∞ into 0 over 0 or ∞/∞.

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Indeterminate form 0 over 0 — apply L prime Hôpital prime s rule. Differentiate the numerator and the denominator separately (this is not the quotient rule).

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Simplify.

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Substitute x equals 1/t (t → 0 to the power plus ).

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Exponentiate the result.

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So the answer is Limit equals e approximately 2.718281828.

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A limit describes the value a function approaches, not necessarily the value it takes. When substitution gives an indeterminate form, the expression is rewritten (L prime Hôpital prime s rule, a conjugate, or logarithms) until substitution works.

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Numerical values as x grows agree: f(127.18281) equals 2.7076718, f(1271.8281) equals 2.7172139, f(12718.281) equals 2.718175, f(127182.81) equals 2.7182711, which checks out.