作业帮求极限limx→0 tanx-sinx/xtanx∧2
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求极限limx→0 tanx-sinx/xtanx∧2
求极限limx→0 tanx-sinx/xtanx∧2
求极限limx→0 tanx-sinx/xtanx∧2
利用等价无穷小,当x→0时,tanx~x,1-cosx~0.5x²
所以原式=lim【x→0】(tanx-sinx)/x³
=lim【x→0】[tanx(1-cosx)]/x³
=lim【x→0】x*(0.5x²)/x³
=0.5
1/2
上下同除以tanx，得到limx→0 1-cosx/xtanx
1-cosx = 2sin^2(x/2)
tanx=sinx/cosx=2sin(x/2)cos(x/2)/cosx
带入计算
=limx→0 sin(x/2)cosx/(2*x/2)cos(x/2)
=limx→0 1/2cosx*(cosx/2)
=1/2
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