设当0＜x≤1时f﹙x﹚=xˆsinx,对于其他x,f﹙x﹚满足f﹙x﹚+k=2f﹙x+1﹚,求常数k使f﹙x﹚在x=0处连续.

Question

设当0＜x≤1时f﹙x﹚=xˆsinx,对于其他x,f﹙x﹚满足f﹙x﹚+k=2f﹙x+1﹚,求常数k使f﹙x﹚在x=0处连续.

Answer 1

f(x)+k=2f(x+1)
x=0时
f(0)=2f(1)-k=2^sin1-k
lim(x→0+)f(x) = lim(x→0+)x^sinx = lim(x→0+)e^(sinxlnx) = e^(lim(x→0+)sinxlnx)
=e^(lim(x→0+)lnx/(1/sinx))=e^( lim(x→0+)(1/x)/(-cosx/sin²x) )
=e^(lim(x→0+)(-sin²x/xcosx) ) = e^(lim(x→0+)(-2sinxcosx/(cosx-xsinx)) )
=e^0 = 1
若使f(x)在x=0处连续,则
lim(x→0+)f(x) = f(0)
即1 = 2^sin1-k
k = 2^sin1-1