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Let L = lim_{x->0} ( (1+x)^(1/x) / e )^(1/x). The value is L = e^a…
Question
Let L = lim_{x->0} ( (1+x)^(1/x) / e )^(1/x). The value is L = e^a for some rational a = m/n in lowest terms (m an integer, n a positive integer). Find |m| + n.
✓ Verified answer: 3checked by our engine — not a guess
Step-by-step solution
We need the logarithm A = (1/x)·( ln(1+x)/x - 1 ).
Since ln(1+x)/x = 1 - x/2 + x^2/3 - ..., we have ln(1+x)/x - 1 = -x/2 + x^2/3 - ....
Dividing by x: A = -1/2 + x/3 - ....
As x -> 0, A -> -1/2, so L = e^(-1/2).
Thus a = -1/2, giving m = -1, n = 2 and |m| + n = 3.
Final answer3
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