1 Limits of sequences and functions of one variable.
Transkrypt
1 Limits of sequences and functions of one variable.
MATHEMATICS 2 levelling classes 2014/2015 (choice by Agnieszka Badeńska) 1 Limits of sequences and functions of one variable. 1.1 Find limits of the following sequences or prove that they are divergent. √ (a) an = n(−1)n ; (−2)n ; n 5n n3 − 5n + 13 = 2 ; 4n − 7 − 3n3 √ ! = n − n2 − 4n + 7 ; !√ = 3 n3 + 6n − 4 − n ; √ = n n · 7n − 4n + n2 · 3n ; p = n 3n2 + cos(n!); (b) an = (c) an (d) an (e) an (f) an (g) an n arctan (nn ) ; n2 + n + 1 3n−2 6n + 5 (i) an = 6n + 1 (h) an = 1.2 Calculate limits if they exist. x2 − x + 2 arctan x ; x→∞ x − x2 1 − cos x lim ; x→0 x sin x √ x2 + 3x − 1 − x √ ; lim x→∞ x − x2 − 2x + 3 1 lim |x| arctan ; x→0 x tan πx √ ; lim x→1 x − x ! lim 2 ln x − ln(x2 + 2x + 3) ; x→∞ 2x x−1 lim ; x→−∞ x+3 1 1 lim ; − x→1 x − 1 ln x (a) lim (b) (c) (d) (e) (f) (g) (h) (i) lim x2 ln(4x); x→0 2 · 2x − 4x ; x→1 x−1 (k) lim+ xsin x ; (j) lim x→0 (l) lim+ (cot x)tan x . x→π Projekt współfinansowany ze środków Unii Europejskiej w ramach Europejskiego Funduszu Społecznego