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 + →