MAST10006 Calculus 2, Semester 2, 2024
Assignment 1
School of Mathematics and Statistics, The University of Melbourne
Due by: 12pm (midday) Monday 12 August 2024
ˆ Answer all questions. Of these questions, one will be chosen for marking.
ˆ Submit your assignment in Canvas LMS as a single PDF file before the deadline above.
ˆ Marks may be awarded for:
ž Correct use of appropriate mathematical techniques.
ž Accuracy and validity of any calculations or algebraic manipulations.
ž Clear justification or explanation of techniques and rules used.
ž Use of correct mathematical notation and terminology.
ˆ You must explicitly state which limit theorems or techniques you use in your answers
when evaluating limits.
ˆ You must use methods taught in MAST10006 Calculus 2 to solve the assignment questions.
ˆ Give any numerical answers as exact values.
Question 1.
(a) Evaluate lim
x→0
sin(3x) + arctan(x)
x+ 5x3
.
(b) Let f : R→ R be given by
f(x) =

x+ a x > 0
2024 x = 0
sin(3x) + arctan(x)
x+ 5x3
x < 0
where a ∈ R is a constant.
i. Find the value(s) of a for which lim
x→0
f(x) exists, or explain why there are none.
ii. Find the value(s) of a for which f is continuous at x = 0, or explain why there are
none. State and use the definition of continuity in your answer.
Question 2. Evaluate the following limits of sequences, or explain why they do not exist.
(a) lim
n→∞
(
n+ 9
n
)2n
(b) lim
n→∞
11n + sin(12n)
13n! + 14
(c) lim
n→∞
sin2
(
1
n
)
ecos(n
2+n)
End of assignment.