MATH1010G Lecture 3: Definitions and results
Refer to Exercise 4.
A. Null sequences of real numbers.
1. Definition. (Null sequences.)
Let cfw_an
n=0 be an infinite sequence in R. cfw_an n=0 is called a null sequence if for any > 0, there exists some N N

MATH1010G Lecture 15: Definitions and Results
Refer to Exercise 19.
For the statements of the various versions of Taylors Theorem, refer to definitions and results for Lecture 14.
A. Taylor polynomials and remainders, and Taylor series
1. Definition. (T

MATH1010G Exercise 7
Themes of the exercise: Special limits associated with powers, exponentiation and logarithm.
1. Evaluate the limits below.
(a)
1
lim x(e x 1)
(b)
x+
lim (x2 + x + 1) sin(
x+
1
)
x2
(c)
lim x ln(
x+
x+1
)
x
2. Let > 1. We are going to

MATH1010G Exercise 5
Themes of the exercise: (1) Sandwich Rule for infinite sequences. (2) Bounded-Monotone Theorem for infinite
sequences.
1. Apply the Sandwich Rule to show that the limits below exist and determine their respective values:
n3 sin(cos(n

MATH1010G Exercise 4
Themes of the exercise: (1) Revision and beyond: Computation of limit of function at the infinities. (2) Limit of
infinite sequence.
1. Evaluate each of the limits below.
(a)
x(3x 1)(2x + 3)
x+ x(5x 3)(4x + 5)
(d)
(b)
2x4 3x2 + 1
x+

MATH1010G Exercise 3 (Solution)
1. (a) Let f : R R be the function defined by f (x) = x3 for any x R.
Pick any s, t R. Suppose s < t. Then
f (t) f (s)
=
t3 s 3
=
(t s)(t2 + st + s2 )
=
(t s)
s2 + t2 + (s + t)2
>0
2
Then f (s) < f (t).
It follows that f i

MATH1010G Exercise 2 (Solution)
1. Here we only provide the answers. Fill in the appropriate steps for justifying the answers.
(a) The equation of the tangent TP to C at P is y = 2.
The equation of the normal NP to C at P is x = 1.
(b) The equation of the

MATH1010G Exercise 6
Themes of the exercise: (1) Various types of limit of function. (2) Continuity. (3) Methods of computation of limits.
1. Let f : R R be the function defined by
x+2
f (x) =
2
x
if
x<1
if
x=1
if
x>1
(a) Sketch the graph of f .
(b) Wha

MATH1010G Exercise 6 (Solution)
1. (a) This is the graph of f :
y
3
bb
2
b
1
bb
0
2
1
x
(b) f (1) = 2. lim f (x) = lim (x + 2) = 1 + 2 = 3. lim f (x) = lim x = 1 = 1.
x1
x1+
x1
x1+
2. (a) This is the graph of f :
y
2
b
1 b
bb
0
b
bb
1
(b)
b
bb
3
2
3
4
x
i

MATH1010G Exercise 7 (Solution)
1. (a) Provided that x > 0 and x is of large magnitude,
1
x(e x 1) =
e1/x 1
1
1/x
as
x +.
(We have used the continuity of the function f defined near 0, given by
t
e 1
t
f (t) =
1
1
= 0.)
x+ x
if
t 6= 0
if
t=0
and lim
1

MATH1010G Exercise 5 (Solution)
1. (a) Suppose n is suciently large.
We have 1 sin(n) 1.
sin(n)
1
1
.
n
n
n
1
1
We have lim = 0 and lim
= 0.
n
n n
n
sin(n)
By the Sandwich Rule, lim
exists and is of value 0.
n
n
Then
(b) Suppose n is suciently large.
We

MATH1010G Exercise 1 (Solution)
1. Provided that t is suciently small but non-zero, we have
f (x + t) f (x)
t
Then f (x) = lim
t0
=
(x + t + 1)4 (x + 1)4
t
=
(x + 1)4 + 4(x + 1)3 t + 6(x + 1)2 t2 + 4(x + 1)t3 + t4 (x + 1)4
t
=
4(x + 1)3 t + 6(x + 1)2 t2 +

2015 MATH1010
Lecture 24: t-method
Charles Li
1
Substitution t = tan x
Let t = tan x, then we have
dx =
dt
.
1 + t2
1
.
1 + t2
t2
.
sin2 x =
1 + t2
cos2 x =
Example 1.1. Compute
Z
cos2 xdx
.
1 + sin2 x
Answer. Let t = tan x. Then
Z 1 dt
Z
cos2 xdx
1+t2 1+

2015 MATH1010
Lecture 20: Integration by part
Charles Li
Remark: the note is for reference only. It may contain typos.
Read at your own risk.
1
Integration by Parts
Let u(x) and v(x) be differentiable functions. By the product rule,
we have
d
dv
du
(uv) =

2015 MATH1010
Lecture 21: Trigonometric integrals
Charles Li
Remark: the note is for reference only. It may contain typos. Read
at your own risk.
1
Trigonometric formula
cos2 + sin2 = 1,
cos 2 = cos2 sin2 ,
sin 2 = 2 sin cos .
1 + cos 2
cos2 =
,
2
1 cos 2

2015 MATH1010
Lecture 23: Partial fraction decomposition
Charles Li
1
Partial Fraction Decomposition
In this section we investigate the antiderivatives of rational functions. Recall that rational functions are functions of the form f (x) =
p(x)
, where p(

MATH1010G Lecture 14: Definitions and Results
Refer to Exercise 18.
A. Rolles Theorem and the Mean-Value Theorem.
1. Rolles Theorem.
Let a, b R, with a < b, and h be a function whose domain contains the whole of the interval [a, b].
Suppose h satisfies a

MATH1010G Lecture 6: Handout on definitions and results
Refer to Exercise 7.
A. Transformaton of limits.
1. Theorem. (Translation of the point where limit is taken.)
right
Let f be a function defined to the
of c, but not necessarily at c.
left
(a)
(b)
(

MATH1010G Lecture 4: Definitions and results
Refer to Exercise 5.
A. Sandwich Rule for limits of infinite sequences.
1. Theorem (Inequality of limits).
Let cfw_an
n=0 , cfw_bn n=0 be infinite sequences in R. Suppose that an bn for any n N. Further suppo

MATH1010G Lecture 2: Definitions and results
Also refer to Exercise 2, Exercise 3.
For the meaning of the asterisk, refer to the definitions and results for Lecture 1.
A. Extrema and derivatives.
1. Definition. (Relative extrema.)
relative minimum
at c

MATH1010G Lecture 1: Definitions and results
Also refer to Exercise 1.
Some of the items in this Handout are marked with an asterisk. You have been under some exposure of the content of
such an item in school mathematics. This exposure is very often at

MATH1010G Lecture 13: Definitions and Results
Refer to Exercise 16, Exercise 17.
A. Two further results on limits and continuity.
1. Lemma. (Inequalities of limits of functions at finite points.)
p.
h(p) > 0
h(p) < 0
greater than
Let p R. Suppose g, h ar

MATH1010G Lecture 7: Handout on definitions and results
Refer to Exercise 8, Exercise 9.
Refer to the Handout on definitions and results for Lecture 5, on the definitions and results for one-sided and two-sided
limits of functions at finite points. The

MATH1010G Lecture 8: Definitions and results
Refer to Exercise 10.
A. Relation between two-sided limit and one-sided limits at a finite point
1. Theorem.
Let f be a function defined near a, but not necessarily at a. The following statements are logically

MATH1010G Handout: Lecture 10
Refer to Exercise 12.
Refer to the definitions and results for Lecture 9, for the meaning of interior of an interval, being defined on an
interval, being continuous (or differentiatiable or . or smooth) on an interval.
A. I

MATH1010G Handout: Lecture 12
Refer to Exercise 14, Exercise 15.
A. Rolles Theorem and its relatives.
1. Rolles Theorem.
Let a, b R, with a < b, and h be a function whose domain contains the whole of the interval [a, b].
Suppose h satisfies all the condi

MATH1010G Handout: Lecture 11
Refer to Exercise 13.
A. LH
opitals Rule for limits of the type 0/0.
1. Theorem. (LH
opital Rule for 0/0 at finite points.)
Let c R, and f, g be functions defined on some open interval interval I with c as its
o
n
greater th

MATH1010G Lecture 18: Definitions and results
Refer to Exercise 22, Exercise 23.
A. Indefinite integration.
1. Mean-Value Theorem for differential calculus.
Let a, b R, with a < b, and F be a function whose domain contains the whole of the interval [a, b

MATH1010G Lecture 9: Definitions and results
Refer to Exercise 11.
A. Notations and terminologies.
1. Definition. (Interior of an interval.)
Let I be an interval. We denote the interval with the endpoints of I removed by I . Each point of I are called an

2015 MATH1010
Lecture 22: Trigonometric substitution
Charles Li
1
More Integrals involving trigonometric functions
Z
Example 1.1. Evaluate
tan x dx.
Answer. Rewrite tan x as sin x/ cos x. While the presence of a
composition of functions may not be immedia