Tiêu đề 08. DIFFERENTIAL CALCULUS (Q1).pdf ✅

DIFFERENTIAL CALCULUS (P – 1) 8.1 Limits 1. limn→∞ (1 2−1)(n−1)+(2 2−2)(n−2)+⋯+((n−1) 2−(n−1))⋅1 (1 3+2 3+⋯+n3)−(1 2+2 2+⋯.+n2) is equal to : (a) 3 4 (b) 1 3 (c) 1 2 (d) 2 3 (6 April 2 nd Shift 2024) 2. limx→0 e−(1+2x) 1 2x x is equal to (a) e (b) 0 (c) −2 e (d) e − e 2 (9 April 2 nd Shift 2024) 3. If a = limx→0 √1+√1+x 4−√2 x 4 and b = limx→0 sin2 x √2−√1+cosx , then the value of ab 3 is : (a) 25 (b) 36 (c) 32 (d) 30 (27 th Jan 1 st Shift 2024) 4. If limx→0 3+αsinx+βcos x+loge(1−x) 3tan2 x = 1 3 , then 2α − β is equal to (a) 5 (b) 1 (c) 7 (d) 2 (27 th Jan 2 nd Shift 2024) 5. limx→0 e 2|sinx|−2|sinx|−1 x 2 (a) does not exist (b) is equal to -1 (c) is equal to 1 (d) is equal to 2 (31 Jan 1 st Shift 2024) 6. Let a1, a2, a3, ... , an be n positive consecutive terms of an arithmetic progression. If d > 0 is its common difference, then limn→∞√ d n ( 1 √a1+√a2 + 1 √a2+√a3 + ⋯ . + 1 √an−1+√an ) is (a) 0 (b) √d