设f(x)=(1-X)tan兀x/2,X∈(0,1],补充定义f(1)=___时,可使f(X)在[0,1]上连续.

题目

设f(x)=(1-X)tan兀x/2,X∈(0,1],补充定义f(1)=___时,可使f(X)在[0,1]上连续.

答案

即需在x=1左连续
即
f(1)=lim x->1- f(x)
=lim x->1- (1-x)tan(pi*x/2)
=lim x->1- (1-x)/cot(pi*x/2)
0/0型,洛必达
=lim x->1- (-1)/(-csc^2(pi*x/2)*pi/2)
=2/pi