求极限这个分子是怎么化简的
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let
y= x-1
y->0
sin(πy) = πy +o(y)
ln(1+y ) = y + o(y)
lim(x->1) [sin(πx) ]^2/ [(x-1)lnx]
=lim(y->0) [sin(π(y+1)) ]^2/ [yln(1+y) ]
=lim(y->0) [sin(π+πy) ]^2/ [yln(1+y) ]
=lim(y->0) [sin(πy) ]^2/ [yln(1+y) ]
=lim(y->0) π^2.y^2/ y^2
=π^2
y= x-1
y->0
sin(πy) = πy +o(y)
ln(1+y ) = y + o(y)
lim(x->1) [sin(πx) ]^2/ [(x-1)lnx]
=lim(y->0) [sin(π(y+1)) ]^2/ [yln(1+y) ]
=lim(y->0) [sin(π+πy) ]^2/ [yln(1+y) ]
=lim(y->0) [sin(πy) ]^2/ [yln(1+y) ]
=lim(y->0) π^2.y^2/ y^2
=π^2
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