《高等数学》课程电子教案（PPT课件）Chapter 1 Functions and Limits §1.6 Limit Theorems

Chapter1FunctionsLimitsand$ 1.6Limit Theorems

Chapter 1 Functions and Limits §1.6 Limit Theorems

Theorem1The sum of finiteinfinitesimalsis also an infinitesimalIf α→0,β→0 as x→x,then α+β→0 as x→xoS1.6LimitTheorem

§1.6 Limit Theorem Theorem 1 The sum of finite infinitesimals is also an infinitesimal. 0, 0 , 0 . → → x → x0 + → as x → x0 If   as then  

Theorem2The product of an infinitesimal and a bounded function isalso an infinitesimalIf α →0 as x →x, when 0<|x- x <8,|f(x)≤ Mthen α f(x)→0 asx→xo:1sinxlim xsinlimForinstancex→0xxx-0Corollary11.The product of an infinitesimal and a constant isalsoaninfinitesimal.2.The product of finite infinitesimals is also an infinitesimal.S1.6LimitTheorem

§1.6 Limit Theorem Theorem 2 The product of an infinitesimal and a bounded function is also an infinitesimal. ( ) 0 . 0 , 0 ( ) 0 0 0 f x x x x x x x f x M → → → →  −   then as If as when ,    For instance x x x 1 lim sin →0 x x x sin lim → 1.The product of an infinitesimal and a constant is also an infinitesimal. Corollary 1 2.The product of finite infinitesimals is also an infinitesimal

Theorem3If lim f(x) = A, lim g(x)= B, thenlim[ f(x)± g(x)) = lim f(x)± lim g(x) = A±Bliml f(x): g(x) = lim f(x)· lim g(x) = A. BAf(x)lim f(x)lim(B±0)Blim g(x)g(x)Corollary2lim kf(x) = k lim f(x) = kAlim[ f(x)]" =[lim f(x)]" = A"lim ^/ f(x) = r/lim f(x) = "/ A(A>0)S1.6LimitTheorem

§1.6 Limit Theorem Theorem 3 If lim f (x) = A, lim g(x) = B, then lim[ f (x) g(x)] = lim f (x) lim g(x) = A B lim[ f (x) g(x)] = lim f (x)lim g(x) = A B ( 0) lim ( ) lim ( ) ( ) ( ) lim = = B  B A g x f x g x f x Corollary 2 lim kf (x) = k lim f (x) = kA n n n lim[ f (x)] = [lim f (x)] = A lim f (x) = lim f (x) = A(A 0) n n n

Example1x2 +9x-5lim(5x2 - 4x)limlimx-1 x+1x-3x→4xApolynomial functionhastheformf(x)= Pm(x)=a,x" +a,xn-1 +...+an-ix+anA rational function is the quotient of two polynomial functionPm(x)aox" +a,xm-l +...+am-ix+amf(x)=Qn(x)box" +bxn-1 +...+bn-ix+b.Theorem4(SubstitutionTheorem)Iff isa polynomial or a rational function,andf(x.)isdefined, then lim f(x)= f(xo)x-→xaS1.6 Limit Theorem

§1.6 Limit Theorem Example 1 lim(5 4 ) 2 3 x x x − → 1 5 lim 1 + − → x x x A polynomial function has the form n n n n f x = Pm x = a x + a x + + a − x + a − 1 1 0 1 ( ) ( )  A rational function is the quotient of two polynomial function n n n n m m m m n m b x b x b x b a x a x a x a Q x P x f x + + + + + + + + = = − − − − 1 1 0 1 1 1 0 1 ( ) ( ) ( )   Theorem 4 (Substitution Theorem) lim ( ) ( ) 0 0 f x f x x x = → If f is a polynomial or a rational function, and is defined, then ( ) x0 f x x x 9 lim 2 4 + →

Example25x2-4x+25x4 -4x3 + 25x°-4x5+60xlimlimlimx→003x2-3x+1x→3x7-3x5+x70x-3x+9x-→0Generally,whena,±0,b,0ao,m=nbom +a,x"-1 +...+am--x+amPm(x)aoxmmlim0.limm0x-→0,m> nS1.6 Limit Theorem

§1.6 Limit Theorem Example 2 3 3 1 5 4 2 lim 2 2 − + − + → x x x x x 7 5 2 4 3 3 3 5 4 2 lim x x x x x x − + − + → 70 3 9 5 4 60 lim 8 6 9 5 − + − + → x x x x x x             = = + + + + + + + + = − − − − → → m n m n m n b a b x b x b x b a x a x a x a Q x P x n n n n m m m m x n m x , 0, , lim ( ) ( ) lim 0 0 1 1 0 1 1 1 0 1   Generally, when 0, 0, a0  b0 

Theorem5 Composite LimitTheoremLet fIg(x)] be composited by u = g(x) and y = f(u) .If lim g(x) = uo, lim f(u) = A, and there existsS, > 0 ,x-→xou>uosuch that g(x)+ u, when x eU(xo,S.), thenlim f[g(x)] = lim f(u) = Ax-→xou-→>uosinxsinxlimForinstanceu= g(x)uxx→>00xsinxsinxSincelim0,, lim u2? = 0, then=0limX-→00xu-→0x->8YS1.6Limit Theorem

§1.6 Limit Theorem Theorem 5 Composite Limit Theorem f g x f u A x x u u = = → → lim [ ( )] lim ( ) 0 0 Let f[g(x)]be composited by u = g(x) and y = f (u). If lim ( ) , lim ( ) , 0 0 g x u0 f u A x x u u = = → → and there exists 0 ,  0  such that g(x)  u0 when xU(x0 , 0 ),then 0, sin Since lim = → x x x For instance , ( ) , sin ( ) 2 f u u x x u = g x = = ? sin lim 2  =      → x x x lim 0, 2 0 = → u u 0 sin then lim 2  =      → x x x

Exerciseslim(n-→00x + arctan x2. limx-→o x -arctan xS1.6 Limit Theorem

§1.6 Limit Theorem ) 1 ) (1 3 1 )(1 2 1 1. lim(1 2 2 2 n n − − − →  x x x x x arctan arctan 2. lim − + → Exercises