羅必達法則 (L’Hospital’s Rule) 1.不定式 (Indeterminate Forms) 2.羅必達定理(L’Hopital’s Rule) 3. 例題 page 659-663 1 Indeterminate Forms 0 0 1. The Indeterminate Forms of Type 2. The Indeterminate Forms of Type 3. The Indeterminate Forms 0 and 4. The Indeterminate Forms 00 , 1 and 0 EX: lim x 0 2 1 cos x x , , lim 2 x x 3 x x e lim x 0 ( x 1)cos x , lim x ln x sin(3x) , lim x x 0 x The Indeterminate Forms of Type 0/0 x2 1 Take lim for example x 1 x 1 2 x 1 0 & When x 1 lim x1 limx 1 0 x1 x2 1 0 lim x1 x 1 0 Divide both numerator and denominator by x-1 x2 1 2 lim x 1 x 1 3 x 1 lim 1 x 1 x 1 The Indeterminate Forms of Type 0/0 x2 1 lim lim x 1 x 1 x 1 x2 1 x 1 x 1 x 1 x2 1 lim x 1 2 x 1 x 1 1 lim x 1 x 1 x2 1 2 2 lim ( x 1)| 2 x x 1 2 x 1 x 1 x 1 x 0 2 lim x 1 x 1 ( x 1) x 0 1 x 1 1 x 1 lim x 1 x 1 4 The Indeterminate Forms of Type 0/0 Replace x 2 1 by f ( x) f (u), f (u) 0 Replace x 1 by g ( x) g (u), g (u) 0 Replace x −1 by x u if f x f (u ) g x g (u ) lim f x lim g x lim ,lim x , x , xu x u xu x u exist and g (u) 0 , then the weak form of L’Hopital’s Rule lim x u 5 f ( x) f ' ( x) g ( x) g ' ( x) L’Hospital’s Rule Let f and g be functions and let a be a real number such that lim f x 0,lim g x 0 xa xa Let f and g have derivative that exist at each point fin' xsome open interval containing a f x lim L lim L g ' x If , then g x f ' x f ' x lim If does exist because g ' x g ' x becomes large without bound for values of x near a, then lim f x also does not exist xa x a x a x u 6 g (u ) EX1 L’Hospital’s Rule x2 1 lim x 1 x 1 Find Check the conditions of L’Hospital’s Rule 2 x lim 1 0 x1 limx 1 0 x1 x2 1 0 lim x 1 0 x 1 If f x x 2 1 then f’(x)=2x If f(x)=x-1 then f’(x)=1 By L’Hospital’s Rule, this result is the desired limit: x 2 1 ' x2 1 2x lim lim 2 lim x 1 1 x 1 x 1 x 1 ' x 1 7 EX2 L’Hospital’s Rule ln x 2 Find lim x 1 x 1 Check the conditions of L’Hospital’s Rule lim ln x 0 x1 If If lim x 1 x1 2 0 ln x lim x 1 x1 2 0 0 then f’(x)= x 1 f(x)= x 12 then f’(x)=2(x-1) f x ln x x 1 Because lim does not exist x1 2 x 1 ln x Then does not exist 2 lim x 1 x 1 8 Using L’Hospital’s Rule f x g x leads to the 1. Be sure that lim xa indeterminate form 0/0. 2. Take the derivates of f and g seperately. f ' x g ' x ; this limit, if it 3. Find the limit of lim xa exists, equals the limit of f(x)/g(x). 4. If necessary, apply L’Hospital’s rule more than once. 9 EX3 L’Hospital’s Rule x3 lim x x 0 e 1 Find Check the conditions of L’Hospital’s Rule limx 3 lim e 0 x 1 x 1 If If f x x3 f(x)= e x 1 2 lim3x 2 x 1 0 then f’(x)= 3x 2 then f’(x)= e x 3x 0 x0 0 lim x x lim e 1 x 0 e x0 10 x3 0 lim x 0 x 1 e 1 x3 0 lim x x 1 e 1 EX4-1 L’Hospital’s Rule ex x x lim x2 x 0 Find lim e x x 0 x 1 0 lim x0 x e x 0 lim x2 1 0 x 0 x 0 2 If f x e x x 1 then f’(x)= e x 1 If f(x)= x 2 then f’(x)=2x ex 1 0 lim 2x 0 x 0 11 EX4-2 L’Hospital’s Rule ex x 1 ex 1 0 lim lim 2 x 0 x 2 x 0 x 0 x f x e 1 If If f(x)= 2x then f’(x)= e x then f’(x)=2 ex 1 lim 2 x 0 2 ex x 1 ex 1 ex 1 lim lim lim 2 x 0 x 0 2 x 2x 2 x 0 12 EX5 L’Hospital’s Rule x2 1 lim x x 1 Find lim x x 1 2 1 0 ex x 1 0 0 lim 1 0 x x 1 x 1 lim x 1 x2 1 2x 1 lim lim 12 1 x x1 x1 2 x x 1 lim x x 1 2 lim x x 1 lim x 1 13 2 1 x 0 0 1 (by substitution) Proof of L’Hospital’s Rule-1 We can prove the theorem for special case f, g, f’, g’ are continuous on some open interval containing a, and g’(a)=0. With these assumptions the fact that lim f x 0 and lim g x 0 xa xa means that both f(a)=0 and g(a)=0 14 Proof of L’Hospital’s Rule-2 Thus, f x f x f (a ) lim lim x a g ( x ) x a g ( x ) g ( a ) Multiplying the numerator and denominator by 1/(x-a) gives f x f (a ) f x xa lim lim x a g ( x ) x a g ( x ) g ( a ) xa 15 Proof of L’Hospital’s Rule-3 By the property of limits, this becomes, f x f (a ) f x lim x a xa lim x a g ( x ) g ( x) g (a) lim x a xa the limit of numerator is f’(a) the limit of denominator is g’(a) and g (a) 0 f ' x f ' a lim f ' x x a lim g '(a) lim g '( x) xa g '( x) x a 16 Proof of L’Hospital’s Rule-4 Thus, f x f ' x f '' x lim lim lim x a g ( x ) x a g '( x ) x a g ''( x ) 17 sin x x .(0/0) Example: Find lim 3 x 0 x lim(sin x 1) 0, lim x 0 3 x 0 x 0 sin x x cos x 1 sin x lim lim lim 3 2 x 0 x 0 x 0 x 3x 6x cos x 1 lim x 0 6 6 18 1 cos x Example: Find lim (0/0) x 0 x 2 3 x lim(1 cos x) 0, lim x 2 3x 0 x 0 x 0 1 cos x sin x lim 2 lim 0 x 0 x 3 x x 0 2 x 3 19 Example: Find e x ln(1 x) 1 lim x 0 x2 (0/0) lim(e x ln(1 x) 1) 0, lim x 2 0 x 0 x 0 1 e e x ln(1 x) 1 1 x lim lim x 0 x 0 x2 2x 1 x e (1 x) 2 lim 1 x 0 2 x 20 The Indeterminate Forms of Type If g ( x) lim f ( x) and lim x u x u Then lim x u 21 f ( x) f ' ( x) lim g ( x) x u g ' ( x ) Example (∞/∞) x 2 x2 Find lim x 3 x 2 5 x lim( x 2 x 2 ) , lim 3x 2 5 x x 0 x 0 x 2 x2 1 2x lim 2 lim x 3 x 5 x x 6 x 5 2 1 lim x 6 3 22 p x Example: Find lim x , where p>0。 x e x p lim x , lim e x x xp px p 1 lim x lim x x e x e lim px ( p 1) 1 p k 0 x k N xp px p 1 lim x lim x x e x e p( p 1) ( p k 1) x p k lim 0 x x e 23 Example: Find sec x lim x ( / 2 ) 1 tan x (∞/∞) sec x sec x tan x lim lim x ( / 2 ) 1 tan x x ( / 2 ) sec 2 x lim sin x 1 x ( / 2 ) 24 Example: Find ln x lim x 2 x (∞/∞) ln x 1/ x lim a lim x x x 1 / x 1 lim 0 x x 25 ln x Example: Find lim (a>0) (∞/∞) x x a lim ln x lim x a x x ln x 1/ x 1 lim a lim a 1 lim a 0 x x x ax x ax 26 (ln x ) 2 Example: Find lim x (∞/∞) x 2 lim(ln x )2 lim 2 x x x 1 ( 2 ln x ) 2 ( 2 ln x ) (ln x ) x lim lim lim x x x (ln 2) x 2 x x x 2 (ln 2) 2 2/ x 0 lim x x x (ln 2) [2 (ln 2) x 2 ] 27 ln x Example: Find xlim 0 cot x lim ln x x (∞/∞) lim cot x x 0 2 ln x 1/ x sin x lim lim lim 2 x 0 cot x x 0 csc x x 0 x sin x lim sin x lim x 0 x 0 x 0 1 0 28 The Indeterminate Forms 0 and f x g x 0 To evaluate lim n z Rewrite Or f x 0 f x g x 1 g x 0 g x f x g x 1 f x Then apply L’Hospital’s Rule 29 The Indeterminate Forms 0 and f x g x To evaluate lim n z F(x)-g(x) must rewrite as a single term. When the trigonometric functions are involved, switching to all sines and cosins may help. 30 1/ 2 x ln x Example: Find xlim 0 lim x1 / 2 0 x 0 1/ 2 lim x x 0 31 lim ln x x 0 0 ln x ln x lim 1/ 2 x 0 x 1/ x lim lim 2 x 0 1 3 / 2 x0 x 0 x 2 1 Example: Find lim x sin 0 x x 1 sin 1 x lim x sin lim x x x 1 x sin t lim t 0 t 1 32 lim (tanx ln sin x ) 0 Example: Find x ( ) 2 lim tan x x ( 2 ) lim ln sin x 0 x( 2 ) ln sin x 0 lim (tan x ln sin x ) lim x ( 2 ) x ( 2 ) cot x 0 1 cos x sin x lim ( cos x sin x ) 0 lim x( 2 ) x ( 2 ) csc2 x 33 x 1 Example: Find lim ( ) (∞−∞) x 1 x 1 ln x x lim x 1 x 1 1 lim x 1 ln x x 1 x ln x x 1 lim( ) lim x 1 x 1 ln x x1 ( x 1) ln x 00 x ln x ln x x 1 / x 1 lim lim x 1 x ln x x 1 x 1 ln x ( x 1) 1 / x ln x 1 1 lim x 1 ln x 2 2 34 1 1 ( ) (∞−∞) Example: Find lim x 1 ln x x 1 1 1 x 1 ln x 0 lim( ) lim x 1 ln x x 1 x 1 ( x 1) ln x 0 1 1/ x x 1 lim x 1 x 1 lim x 1 x ln x x 1 ln x x 1 1 lim x 1 2 ln x 2 35 Example: Find lim (sec x tan x ) (∞−∞) x ( 2 ) lim sec x x ( 2 ) lim tan x x ( 2 ) 1 sin x lim (sec x tan x) lim x ( 2 ) x ( 2 ) cos x cos x lim 0 x ( 2 ) sin x 36 00 [ln 2 x ln( x 1)] Example: Find lim x lim ln 2 x x lim ln( x 1) x 2x lim[ln 2 x ln( x 1)] lim ln x x x 1 2x ln( lim ) ln( lim 2 ) ln 2 x x 1 x 1 37 1 0 Example: Find lim x tan x x 1 tan 1 x 0 lim x tan lim 0 x x 1 x x 2 sec t tan t lim 1 lim t 0 t 0 1 t 38 Example: Find lim( x ) tan x 0 x lim ( x 2 2 2 2 x ) tan x lim ( x 2 2 x ) tan[( x 2 ) lim( t )( cot t ) lim( t ) tan( t ) t 0 2 t 0 t cos t t cos t t lim lim lim lim cos t t 0 sin t t 0 sin t t 0 sin t t 0 1 lim lim cos t 1 1 1 t 0 cos t t 0 39 2 ] The Indeterminate Forms 0 0 0 , and 1 In these cases g x y f x 1. Let 2. ln y g x ln f x g x ln f x exists and equal L, 3. If lim x a g x then lim f x e L x a 40 Example: Find lim (1 x ) 1 x 0 lim (1 x ) x 0 lim (1 x )cot x x 0 lim cot x 0 cot x lim e lim (1 x)cot x 1 x 0 cot x ln(1 x ) x 0 ln(1 x ) exp[ lim ] x 0 tan x 1 exp[lim ]1 2 x 0 (`1 x ) sec x 41 n lim Example: Find n n let y n n ln n ln y n 1 ln x lim ln y lim lim x 0 n x x n 1 and lim ln y ln( lim y ) 0 lim y 1 n Then 42 n lim n n 1 n n 1 x (1 ) Example: Find lim x x 1 1 lim(1 ) 1 lim x x x x 1 x ln(1 ) 1 x x lim(1 ) lim e x x x 1 exp[ lim x ln(1 )] x x 1 lim x ln(1 ) 0 x x 43 1 ln(1 ) 1 x lim x ln(1 ) lim x 1 x x x ln(1 t ) lim t 0 t Then 44 1 x lim(1 ) exp( 1) e x x 00 1 lim 1 t 1 t 0 1 cot x lim ( 1 sin 4 x ) Example: x0 。 lim (1 x ) 1 lim cot x 0 1 x 0 lim (1 x )cot x lim ecot x ln(1sin 4 x ) x 0 x 0 ln(1 sin 4 x ) exp[ lim ] x 0 tan x 4 cos4 x 4 exp[lim ] e 2 x 0 (`1 sin 4 x ) sec x 45 x 0 lim x Example: Find 0 x 0 x ln x ) lim x x lim e x ln x exp( xlim 0 x 0 x 0 1 ln x lim x lim ( x ) 0 lim x ln x lim 1 x 0 x 0 x0 x0 1 2 x x lim x exp( lim x ln x) exp(0) 1 x x 0 46 x 0 1 x 0 Example: Find lim ( 2 ) x 0 x 1 x lim( 2 ) lim e x 0 x x 0 x ln 1 x2 1 exp( lim x ln 2 ) x0 x 1 x ln 2 Replace the result of lim x 0 x 47 1 ln 2 1 lim x ln 2 lim x x 0 x 0 1 x x 2 2 x 3 x lim x 0 1 2 x lim 2 x 0 x 0 1 x 1 Then lim( 2 ) exp( lim x ln 2 ) exp( 0) 1 x 0 x x 0 x 48 Example: lim(e x ) x 1/ x x 0 lim(e x) x x 0 1/ x 1 1 / x ln( e x x ) lim e x0 ln(e x ) lim exp( ) x 0 x ln(e x x ) exp(lim ) x 0 x x 49 (e x ) Example: lim x 0 x 1/ x e 1 and x x ln(e x ) lim lim e x 1 x 0 x 0 x 1 x Then lim(e x x )1/ x x 0 ln(e x x ) exp(lim ) x 0 x exp(1) e 50