Tanx-sinx/sinx的立方的极限 如图
sinx^tanx\u7684\u6781\u9650\u7b54\u6848\u662f( sinx)^tanx=1
\u5177\u4f53\u6b65\u9aa4\u5982\u4e0b\uff1a
ln lim (x\u21920) ( sinx)^tanx
=lim (x\u21920) ln(sinx)^tanx
=lim (x\u21920) tanx*ln(sinx)
=lim (x\u21920) ln(sinx)/cotx
=lim (x\u21920) (cosx/sinx)/(-1/sin²x)
=lim (x\u21920) -(cosxsinx)
=0
\u5219lim (x\u21920) ( sinx)^tanx=1
\u6269\u5c55\u8d44\u6599\u6781\u9650\u7684\u6c42\u6cd5\u6709\u5f88\u591a\u79cd\uff1a
1\u3001\u8fde\u7eed\u521d\u7b49\u51fd\u6570\uff0c\u5728\u5b9a\u4e49\u57df\u8303\u56f4\u5185\u6c42\u6781\u9650\uff0c\u53ef\u4ee5\u5c06\u8be5\u70b9\u76f4\u63a5\u4ee3\u5165\u5f97\u6781\u9650\u503c\uff0c\u56e0\u4e3a\u8fde\u7eed\u51fd\u6570\u7684\u6781\u9650\u503c\u5c31\u7b49\u4e8e\u5728\u8be5\u70b9\u7684\u51fd\u6570\u503c
2\u3001\u5229\u7528\u6052\u7b49\u53d8\u5f62\u6d88\u53bb\u96f6\u56e0\u5b50\uff08\u9488\u5bf9\u4e8e0/0\u578b\uff09
3\u3001\u5229\u7528\u65e0\u7a77\u5927\u4e0e\u65e0\u7a77\u5c0f\u7684\u5173\u7cfb\u6c42\u6781\u9650
4\u3001\u5229\u7528\u65e0\u7a77\u5c0f\u7684\u6027\u8d28\u6c42\u6781\u9650
5\u3001\u5229\u7528\u7b49\u4ef7\u65e0\u7a77\u5c0f\u66ff\u6362\u6c42\u6781\u9650\uff0c\u53ef\u4ee5\u5c06\u539f\u5f0f\u5316\u7b80\u8ba1\u7b97
lim(tanx-sinx)/(sinx)^3
=lim(sinx-sinxcosx)/(sinx)^3cosx
=lim(1-cosx)/(sinx)^2cosx
=lim(1/2)x^2/(sinx)^2cosx
=lim(1/2)x^2/(x^2cosx)
=(1/2)lim1/cosx
=1/2
因为tanx-sinx=sinx(1-cosx)/cosx，当x→0时，1-cosx～(1/2)x^2，sinx～x，所以，用等价无穷小替换，就可以求得极限为1/2.
1/2
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