1.∵f(x)=cos(πx/2)/[x²(x-1)]
∴它的間斷點(diǎn)是：x=0,x=1
∵f(0+0)和f(0-0)不存在
f(1+0)=f(1-0)=lim(x->1){cos(πx/2)/[x²(x-1)]}
=lim(x->1)(1/x²)*lim(x->1)[cos(πx/2)/(x-1)]
=1*(-π/2)
=-π/2
∴x=0是屬于第二類間斷點(diǎn)
x=1是屬于可去間斷點(diǎn)
在原函數(shù)中,令x=1時(shí),y=-π/2
原函數(shù)在點(diǎn)x=1就連續(xù)了
2.∵y=(x²-2x)/[|x|(x²-4)]
∴它的間斷點(diǎn)是：x=0,x=2,x=-2
∵f(0+0)=1/2,f(0-0)=-1/2
f(2+0)=f(2-0)=1/4
f(-2+0)和f(-2-0)都不存在
∴x=0是屬于第一類間斷點(diǎn),x=-2是屬于第二類間斷點(diǎn)
x=2是屬于可去間斷點(diǎn),補(bǔ)充定義：當(dāng)x=2時(shí),y=1/4.原函數(shù)在點(diǎn)x=2就連續(xù)了
3.∵y=x/tanx
∴x=kπ,x=kπ+π/2 (K是整數(shù))是它的間斷點(diǎn)
∵f(0+0)=f(0-0)=1 (K=0時(shí)）
f(kπ+0)和f(kπ-0)都不存在 （k≠0時(shí)）
f(kπ+π/2+0)=f(kπ+π/2-0)=0
∴x=kπ (是不為零的整數(shù)）是屬于第二類間斷點(diǎn),
x=0和x=kπ+π/2 (K是整數(shù))是屬于可去間斷點(diǎn)
補(bǔ)充定義：當(dāng)x=0時(shí),y=1.當(dāng)x=kπ+π/2 (K是整數(shù))時(shí),y=0.
原函數(shù)在點(diǎn)x=0和x=kπ+π/2 (K是整數(shù))就連續(xù)了.
4.∵F(0+0)=F(0-0)=lim(x->0){[cos(2x)-cos(3x)]/x²}
=lim(x->0){[-2sin(2x)+3sin(3x)]/(2x)} (利用羅比達(dá)法則)
=lim(x->0){[(-2)sin(2x)/(2x)][(9/2)sin(3x)/(3x)]}
=lim(x->0)[(-2)sin(2x)/(2x)]*lim(x->0)[(9/2)sin(3x)/(3x)]
=(-2)*(9/2) (利用特殊極限lim(x->0)(sinx/x)=1)
=-9
∴當(dāng)a=-9時(shí),函數(shù)F(x)在x=0處連續(xù).