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Let L = lim_{x->0} ( cosh(x)·cos(x) - 1 ) / x^4. Writing L = m/n in…
Question
Let L = lim_{x->0} ( cosh(x)·cos(x) - 1 ) / x^4. Writing L = m/n in lowest terms (m an integer, n a positive integer), find |m| + n.
✓ Verified answer: 7checked by our engine — not a guess
Step-by-step solution
Use cosh(x) = 1 + x^2/2 + x^4/24 + ... and cos(x) = 1 - x^2/2 + x^4/24 - ....
Multiply: cosh(x)cos(x) = (1 + x^2/2 + x^4/24)(1 - x^2/2 + x^4/24) + O(x^6). The product = 1 + (x^2/2 - x^2/2) + (x^4/24 + x^4/24 - x^4/4) + O(x^6) = 1 + (1/24 + 1/24 - 1/4)x^4 + O(x^6) = 1 - (1/6)x^4 + O(x^6).
So cosh(x)cos(x) - 1 = -(1/6)x^4 + ..., and dividing by x^4 gives L = -1/6.
Thus m = -1, n = 6, |m| + n = 7.
Final answer7
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