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Let f(x) = lim_(n->inf) (e^x + e^(-x) * x²ⁿ) / (e^x * x²ⁿ + e^(-x)). Which of the following is correct?

1. f(1⁺) = e²
2. f(1⁻) = 1/e²
3. f(1) = e²
4. f(1⁺) = 1/e²

Correct answer: f(1⁺) = 1/e²

Solution

Case 1: |x|>1 (x->1+): x²ⁿ->inf; divide numerator and denominator by x²ⁿ: f = (e^x/x²ⁿ+e^(-x))/(e^x+e^(-x)/x²ⁿ) -> e^(-x)/e^x = e^(-2x). At x->1+: f(1+) = e⁻² = 1/e². Case 2: |x|<1 (x->1-): x²ⁿ->0; f = e^x/e^(-x) = e^(2x). At x->1-: f(1-) = e². Case 3: x=1 exactly: x²ⁿ=1; f=(e+e⁻¹)/(e+e⁻¹)=1.

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