lim x(π/2-arctanx)=?lim下面是+∞
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lim x(π/2-arctanx)=?lim下面是+∞
方法1
lim[x(π/2-arctanx)]
=lim(π/2-arctanx)/(1/x) (0/0型,用洛必达法则)
=lim(1/(1+x^2))/(1/x^2)
=limx^2/(1+x^2)
=lim[1-1/(1+x^2)]
=1
方法2
设arctanx=t.则x=tant
lim(x→∞)x(π/2-arctanx)
=lim(t→π/2)tant(π/2-t)
=lim(t→π/2)sint*[(π/2-t)/sin(π/2-t)]
=1
((π/2-t)/sin(π/2-t)->1,
lim[x(π/2-arctanx)]
=lim(π/2-arctanx)/(1/x) (0/0型,用洛必达法则)
=lim(1/(1+x^2))/(1/x^2)
=limx^2/(1+x^2)
=lim[1-1/(1+x^2)]
=1
方法2
设arctanx=t.则x=tant
lim(x→∞)x(π/2-arctanx)
=lim(t→π/2)tant(π/2-t)
=lim(t→π/2)sint*[(π/2-t)/sin(π/2-t)]
=1
((π/2-t)/sin(π/2-t)->1,
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