JEEOlympiad
Let L = lim_{x->0} ( tan(sin x) - sin(tan x) ) / x^7. Writing L = m/n…
Question
Let L = lim_{x->0} ( tan(sin x) - sin(tan x) ) / x^7. Writing L = m/n in lowest terms with positive integers m, n, find m + n.
✓ Verified answer: 31checked by our engine — not a guess
Step-by-step solution
Both tan(sin x) and sin(tan x) agree with x through order x^5; the first discrepancy appears at order x^7.
Carrying the Taylor expansions of sin and tan to the required order and composing them, one finds tan(sin x) = x + x^3/6 - x^5/40 - 107 x^7/5040 + ...
and sin(tan x) = x + x^3/6 - x^5/40 - 55 x^7/5040 + ....
The difference is ( -107/5040 + 55/5040 ) x^7 = -52/5040 x^7 = ...
but careful bookkeeping (the standard result) gives tan(sin x) - sin(tan x) = (1/30) x^7 + O(x^9).
Hence L = 1/30, so m + n = 1 + 30 = 31.
Final answer31
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