TMA2101: Determine the Values of the Following Limits- limx→10 [(x − 8)^3 + e^(x−9)^2+ ln(x − 2)], Calculus for Computing Assignment, NUS, Singapore

TMA2101 Calculus for Computing: Take-Home Test

Question 1

Determine the values of the following limits, whenever the limit
exists.
(a) limx→10 [(x − 8)^3 + e^(x−9)^2+ ln(x − 2)]
(b) limx→2 cos(x − 2) sin(2x − 4)/ (x − 2)
(c) limx→2 sin(x − 2) cos (1/(x − 2)^8)
(d) limx→∞ (x + 2)^6 sin (10/ (x + 2)^6)
(e) limx→−1 (3 − x)^3 / (x − 1)^3 (x + 1)^2
(f) limx→∞ sin (10x + 2/ x − 1 − 4x^2 + 5/ x^3 − 1)^2
(g) limx→1 (sin((x − 1)^2) + sin (1/ (x − 1)^2)
(h) limx→1 x^2 sin ( 1/ x + 1)
(i) limx→∞ (e^3x + e^4x)/ (e^4x + e^x)

TMA2101 Calculus for computing: Take-Home Test

Question 2

see Question 1

Question 3

Find the limits of the following convergent sequences.
(a) an = (e)^(1/n) + (0.1)^(1/n) + (0.2)^n
(b) an = 10 + 10n^3 + sin n/ n^3 + 20n^2 + n^(−3)
(c) an = 3^n + 2(4^n) + e^n / 4^n + 2^n
(d) an = sin^2(n^2 + n + e) / (1.001)^n
(e) an = (n + 1 / n + 2)^n

Question 4

Let a1 = 1, an+1 = 2an + 3/ 4, n = 1, 2, . . ..
Prove, by induction, that
(a) an < 2 for all n,
(b) the sequence (an) is increasing.
Find limn→∞ an.