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Let L = lim_{x->0} ( (1 + x^2)^(1/3) - cos(x) ) / ( x·sin(x) ).…
Question
Let L = lim_{x->0} ( (1 + x^2)^(1/3) - cos(x) ) / ( x·sin(x) ). Writing L = m/n in lowest terms with positive integers m, n, find m + n.
✓ Verified answer: 11checked by our engine — not a guess
Step-by-step solution
Expand: (1+x^2)^(1/3) = 1 + (1/3)x^2 + O(x^4) and cos(x) = 1 - x^2/2 + O(x^4). The numerator = (1/3 + 1/2)x^2 + O(x^4) = (5/6)x^2 + O(x^4).
The denominator x·sin(x) = x·(x - x^3/6 + ...) = x^2 + O(x^4).
So the ratio tends to (5/6) / 1 = 5/6. Thus L = 5/6, m + n = 5 + 6 = 11.
Final answer11
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