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2019/2020 (Semester 2)
TMA2101 Calculus for Computing
Take-Home Test (page 1)
Submit your answer script to submission folder at LumiNUS TMA2101
by Sunday, 1 March 2020, 11.59 pm.
Answer script in PDF format.
Title of your answer script: Your Name.
Write your name on your answer script.
Do All Questions (There are Two Pages).
1. (54 marks) Determine the values of the following limits, whenever the limit
exists.
(a) lim
x→10
h
i
2
(x − 8)3 + e(x−9) + ln(x − 2)
cos(x − 2) sin(2x − 4)
x→2
(x − 2)
1
(c) lim sin(x − 2) cos
x→2
(x − 2)8
10
(d) lim (x + 2)6 sin
x→∞
(x + 2)6
(3 − x)3
(e) lim
x→−1 (x − 1)3 (x + 1)2
2
10x + 2 4x2 + 5
(f) lim sin
− 3
x→∞
x−1
x −1
1
2
(g) lim
sin((x − 1) ) + sin
x→1
(x − 1)2
1
(h) lim x2 sin (
)
x→1
x+1
e3x + e4x
(i) lim
x→∞ e4x + ex
(b) lim
TMA2101 Calculus for computing
Take-Home Test (page 2)
1. see page 1
2. (30 marks) Find the limits of the following convergent sequences.
1
1
(a) an = (e) n + (0.1) n + (0.2)n
10 + 10n3 + sin n
n3 + 20n2 + n−3
3n + 2(4n ) + en
(c) an =
4n + 2n
2
sin (n2 + n + e)
(d) an =
(1.001)n
n
n+1
(e) an =
n+2
(b) an =
3. (16 marks) Let a1 = 1, an+1 =
2an + 3
, n = 1, 2, . . ..
4
Prove, by induction, that
(a) an < 2 for all n,
(b) the sequence (an ) is increasing.
Find lim an .
n→∞
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