7
LIMITS AND CONTINUITY
Karl Weierstrass (1815 – 1897), a German mathematician, first
introduced the delta-epsilon definition of a limit in the form it is usually
written today. He also introduced the notations lim and lim x → xO.
The modern notation of placing the arrow
below the limit symbol, that is, x→x
lim is
due to Godfrey Harold Hardy (1877 –
1947), an English mathematician, in his
book A Course of Pure Mathematics in
1908.
O
Godfrey Harold Hardy
Concept Map
Limits and Continuity
Limits
Continuity
Asymptotes
Continuity of a
function
Intermediate value
theorem
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2
ACE AHEAD Mathematics (T) Second Term
LEARNING OUTCOME
Determine the existence and
values of the left-hand limit,
right-hand limit and the limit
of a function.
7.1 Limits
The Limit of a Function
The limit of a function f(x) at a number x = a refers to the value taken
by f(x) as x approaches a. It is denoted by lim f(x).
x→a
If the function f(x) approaches the limit ᐉ as x approaches a, then it is
written as lim f(x) = ᐉ. This function is illustrated in Figure 7.1.
x→a
y
y = f(x)
lim f(x)
x → a+
ᐉ
lim f(x)
x → a−
Figure 7.1
x
a
O
• If x approaches the number a from the right-hand side, it is written
as lim+ f(x).
x→a
lim f(x) is also known as the right-hand limit.
x→a+
• If x approaches the number a from the left-hand side, it is written
as lim− f(x).
x→a
lim f(x) is also known as the left-hand limit.
x→a−
If lim+ f(x) = lim− f(x) = ᐉ, then the lim f(x) exists and equals ᐉ.
x→a
x→a
x→a
If lim+ f(x) ≠ lim− f(x), then the lim f(x) does not exist.
x→a
x→a
x→a
Let the function f in Figure 7.2 be f(x) = x 2 − 2x + 3 and let x approaches
the number ‘3’.
Table 7.1(a) shows that the function f(x) = x 2 − 2x + 3 approaches
the limit ‘6’ as x approaches the number ‘3’ from the left while Table
7.1(b) shows that the function f(x) = x 2 − 2x + 3 approaches the limit
‘6’ as x approaches the number ‘3’ from the right.
x
2.8
2.9
2.99
2.999
2.9999
2.99999
f(x) = x 2 − 2x + 3
5.2400
5.6100
5.9601
5.9960
5.9996
6.0000
∴ lim− f(x) = 6
x→3
Table 7.1(a)
CH007-Maths.indd 2
x
3.2
3.1
3.01
3.001
3.0001
3.00001
f(x) = x 2 − 2x + 3
6.8400
6.4100
6.0401
6.0040
6.0004
6.0000
∴ lim+ f(x) = 6
x→3
Table 7.1(b)
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20
ACE AHEAD Mathematics (T) Second Term
Checkpoint 7.3
2x + 1
, x ≠ 0.
x
Find lim f(x) and hence, determine the horizontal asymptote of the graph of the function f.
1 Determine the vertical asymptote of the graph of the function f(x) =
x→±∞
Sketch the graph of the function f.
Find f −1. State the domain and the range of f −1.
2x + 3
, x ≠ 1.
1−x
Find lim g(x) and hence, determine the horizontal asymptote of the graph of the function g.
2 Determine the vertical asymptote of the graph of the function g(x) =
x→±∞
Sketch the graph of the function g.
Find g−1. State the domain and the range of g−1.
Summary
1 If the function f(x) approaches the limit ᐉ as x approaches a, then it is denoted by lim f(x) = ᐉ.
x→a
2 If x approaches a number a from the right-hand side, it is written as lim+ f(x).
x→a
3 If x approaches a number a from the left-hand side, it is written as lim− f(x).
4 lim f(x) = ᐉ if and only if lim+ f(x) = lim− f(x) = ᐉ.
x→a
x→a
x→a
x→a
5 The laws of limits are
• lim c f(x) = c lim f(x)
x→a
x→a
• lim [f(x) ± g(x)] = lim f(x) ± lim g(x)
x→a
x→a
x→a
• lim [f(x) ⴢ g(x)] = lim f(x) ⴢ lim g(x)
x→a
x→a
x→a
lim f(x)
• lim
x→a
f(x)
, where lim g(x), ≠ 0
= x→a
g(x) lim g(x)
x→a
x→a
6 A function f is continuous at x = a if the lim f(x) and f(a) both exist and are equal,
i.e. lim f(x) = ᐉ = f(a).
x→a
x→a
7 If lim+ f(x) = ±∞ or lim− f(x) = ±∞, then x = a is a vertical asymptote of f(x).
x→a
x→a
8 If lim f(x) = k, then y = k is a horizontal asymptote of f(x).
x→±∞
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Model Exam (Paper 2)
Time: 1½ hours
Section A [45 marks]
Answer all questions in this section
1 The function g is defined by
g(x) =
{
x−2,
|x − 2|
4,
x + 2,
x<2
x=2
x>2
(a) Without using a graph, determine whether g is continuous at x = 2 or not.
[3 marks]
(b) Sketch the graph of y = g(x).
[2 marks]
2 (a) Find the gradient of the curve y = ln
(b) By using the substitution u2 = x, find
x + y at the point where its y-coordinate is 1.
∫
1
4
0
dx
.
x (1 − x)
[4 marks]
[6 marks]
3 Show that the equation x = −e x has a root between −1 and 0.
[2 marks]
By using the Newton–Raphson method and taking −0.37 as the first estimation, calculate the root
correct to two decimal places.
[6 marks]
2x − 1
4 Sketch the graph of the curve y =
, indicating clearly its asymptote and turning point and
(x
− 2)3
the nature of the turning point.
[9 marks]
5 Sketch the curves y = −e x and y = −2 − 3e−x on the same coordinate axes.
Hence, find the area of the region bounded by the curves and the y-axis.
[2 marks]
[7 marks]
dy
6 Find the particular solution of the differential equation x + 2y = x2 such that x > 0 and y = 2,
dx
when x = 1, expressing y in terms of x.
[4 marks]
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Coursework Assignment
Answer all questions in this section.
1 (a) When a population is not constrained by environment limitation, they tend to grow at a rate
that is proportional to the size of population. By taking y = y(t) as the population with respect
to time t and y0 as initial population, show that y(t) = y0ekt.
(b) According to a Botanical Department data, the population of a type of butterfly in 2017
was approximately 5900 and growing at a rate of 1.33% per year. Assuming an exponential
growth model, estimate the population of the butterfly at the beginning of year 2037.
2 The data obtained from breeding of butterflies in a university by a group of researchers shows
that the relative growth rate is almost constant when the population is small. The relative growth
rate decreases as the population increases and becomes negative if the population exceeds its
carrying capacity L.
(a) Show that this population growth rate is given by
dy
k
= ky − y 2.
dt
t
(b) Explain what will happen when y is very small relative to L.
(c) Comment on the meaning of the values of P for constant growth, increasing growth and
declining growth.
dy
(d) Find the maximum value of . What do you understand about this value?
dt
(e) Suppose y0 is the initial population and y = y(t) at time t, show that
y0L
y=
.
y0 + (L − y0)e−kt
Hence by using different population sizes y0, L and K, plot on the same axis, a few graphs to
show the behaviour of y versus t. Comment on your curve.
3 Let y(t) be the population of a certain insect species, assume y(t) satisfies logistic growth equation
冤
冥
dy
y
= 0.2 y 1 −
, y = y(0) = 150.
dt
200 0
(a) Without solving the differential equation, sketch the graph of y(t).
(b) What is the long term behaviour of y(t)?
(c) Show that the solution in the form of
y(t) =
e0.2t
.
A + Be0.2t
Hence, find A and B.
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