o F(S+a)=Lle f(ol lit atys 例:La=(s+a F 2: Lleo coste(t(S+a)+o2 o e S+a 五初值定理和终值定理L10M/ 2 S+O 初值定理:0在t=0处无冲激则 了(0)=mimf()=limS( S→0 终值定理:imf(t)存在时 oA lim SF(S)=lim f(t=f(oo) →01→0
2 L[e cos t (t)] t 例 ： − 2 2 ( ) + + + = S S 2 ( ) 1 + = S 五.初值定理和终值定理 (0 ) lim ( ) lim ( ) 0 f f t SF S s t→ → + = = + 初值定理： f(t)在t = 0处无冲激则 F(S ) L[e f (t)] t − + = 1 L[te (t)] t 例 ： − lim f (t)存在时 t→ lim ( ) lim ( ) ( ) 0 = = → → SF S f t f s t 2 1 [ ( )] S L t t = 2 2 [cos ] + = s s L t 终值定理：
f(∞)=imf()= lim SF(S) 证:利用导数性质 07 imf()e“=lmSF(S)一f(0川 s→)0 d ro d (lime stat=/(er -F(o)=lim SF(S)-(O) 公 例已知P(38+23+3(0) 3s2+4+5 3(S2+2+3 =im 3
( ) lim ( ) lim ( ) 0 f f t SF S t→ s→ = = 证：利用导数性质 lim ( ) lim[ ( ) (0 )] 0 0 0 − → − → − f t e dt = SF S − f dt d s st s f t e dt dt d st s − → − 0 0 ( )lim (0 ) ( 2 3) 3 4 5 1 ( ) 2 2 + + + + + = f S S S S S 例 ：已知F S 求 3 ( 2 3) 3 4 5 lim 2 2 = + + + + = → S S S S s = − 0 f ((t)) (0 ) lim ( ) (0 ) 0 − → − = f − f = SF S − f s