1 0, !4!3!2 1 ) ≤ +≈ ≤≤ + + ++≈ xxx tgx bxxxxx eax
Transkrypt
1 0, !4!3!2 1 ) ≤ +≈ ≤≤ + + ++≈ xxx tgx bxxxxx eax
LIST 4. The de L’Hospital and Taylor’s Theorems. Task 1: Using the de L’Hospital theorem calculate the limits: π x ; b ) lim+ x→ 0 πx ctg 2 ln(cos 2 x + 1) ; a ) lim − π ln cos x x→ 2 πx 2 ; ln x ln sin e ) lim x→1 f ) lim + x→ 0 chx − 1 ; 1 − cos x g ) lim x→ 0 i ) lim xarcctgx ; j ) lim+ x ln x; k ) lim− (π − x )tg x→∞ x→0 1 n ) lim− − ctgx ; x→ 0 x s ) lim+ ctgx x→0 x →π 1 1 o ) lim − ; x →1 ln x x −1 1 ln x ; ( t ) lim+ 1 + x x →0 ) 2 1/ x ln( 2 x + 1) ; c ) lim x→ ∞ x d ) lim ln x ; x x→ ∞ 3 x − arctgx x 10 − 10 x + 9 ; h ) lim ; x→1 x2 x5 − 5x + 4 1 1 x − 2 ; ; l ) lim− (1 − cos x )ctgx ; m ) lim+ x → 0 1 − cos x x→0 x 2 3 p ) lim+ x 4 + ln x ; q) lim+ (− ln x) x ; x→ 0 ; x→0 r) lim− (tgx) cos x ; π x→ 2 2 w) lim arctgx . x →∞ π u ) lim+ (1 + x ) ; x ln x x →0 Task 2: Write the Taylor formula 4th order for the functions f (x) in the points x0: x a. f ( x ) = e x , x0 = −1; b. f ( x ) = , x 0 = 2. x −1 Task 3: Write the Taylor formula nth order for the functions f (x) in the points x0: a. f ( x ) = 1 , x0 = −1; x b. f ( x ) = x , x 0 = 4. Task 4: Estimate the absolute errors of the approximate formulas: x2 x3 x 4 a) e ≈ 1 + x + + + , 0 ≤ x ≤ 1; 2! 3! 4! x x3 b) tgx ≈ x + , x ≤ 0,1. 3 Task 5: Find approximate values of the numbers below with the absolute error smaller then d. 1 a) 4 , d = 0,01; b) cos10o , d = 0,001. e