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If lim_(x→∞) (1+(λ)/(x)+(μ)/(x²))^(2x)=e², which of the following values of λ and μ is correct?

1. λ=-1,μ=2
2. λ=2,μ=1
3. λ=1,μ can be any real number
4. λ=μ can be any real number

Correct answer: λ=1,μ can be any real number

Solution

As x->infinity, (1+lambda/x+mu/x^2)^(2x) -> exp(2 lambda). Setting e^(2 lambda)=e^2 gives lambda=1, and mu can be any real number.

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