求极限lim(arcsinx*arctanx/2x^2)x趋近于0
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求极限lim(arcsinx*arctanx/2x^2)x趋近于0
先看lim arcsinx/x
设arcsinx=t
x=sint
x趋近于0,t也趋近于0
lim arcsinx/x
=lim t/sint
=1
再看lim arctanx/x
设arctanx=t,
x=tant
x趋近于0,t也趋近于0
cost趋近于1,t/sint趋近于1
lim arctanx/x
=lim t/tant
lim cost*t/sint
=1
所以
lim(arcsinx*arctanx/2x^2)
=(lim arcsinx/x * lim arctanx/x)/2
=1/2
设arcsinx=t
x=sint
x趋近于0,t也趋近于0
lim arcsinx/x
=lim t/sint
=1
再看lim arctanx/x
设arctanx=t,
x=tant
x趋近于0,t也趋近于0
cost趋近于1,t/sint趋近于1
lim arctanx/x
=lim t/tant
lim cost*t/sint
=1
所以
lim(arcsinx*arctanx/2x^2)
=(lim arcsinx/x * lim arctanx/x)/2
=1/2
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