LESSON 26
TOPIC 8: LIMITS AND CONTINUITY
8.1 - Limits
8.2 – Asymptotes
8.3 – Continuity
1
8.1 Limits
Learning outcomes
At the end of this topic, students should be able to :
1
State limit of a function f(x) as x approaches a
given value a, l i m f ( x ) = L
x  a
2
3
State the basic properties of limit.
Find
lim f ( x )
x
a
when
lim g( x )
x
a
x
a
0 and
a
lim g( x )
x
lim f ( x )
0 by the following methods :
i. Factorisation ii. Multiplication of conjugates
For more details, please visit
https://www.mathsisfun.com/calculus/limits.html
2
8.1 LIMITS
Consider f(x) = x2
What is the value of f(x) when x approaches 2 ?
y
4
1.9
From left,
2
2.1
x
From right,
x
f(x)
x
f(x)
1.9
3.610
2.1
4.410
1.99
3.960
2.01
4.040
1.999
3.996
2.001
4.004
3
Notice that when x approaches 2 from the left side or from
the right side, the value of f(x) approaches 4.
Mathematically:
When x  2, f(x)  4
The notation:
lim f(x) = lim(x )
2
x 2
x 2
=
4
Concept of Limit
If the function f approaches a number L when x approaches
a, then we say that the limit of f(x) when x approaches a is L
lim f (x ) = L
x a
y
L
a
x
5
Note:
Limit examine the behaviour of a function f(x) as the value x
gets closer and closer to number of a (but not equal to a).
Intuitively, limits can be evaluated by substituting x = a into
the function
lim f (x ) = f (a )
x a
6
Basic Properties of Limits
1) lim c = c
x a
2)
lim x = a
xa
where c=constant
Examples
Examples
a) lim -1=
a) lim x =
b)
b) lim x =
x5
lim 2 =
x -3
x 2
x -9
7
3) lim x = a
n
xa
n
where n is an integer
Example
lim x3 =
x2
8
4) lim  f(x) ±g(x)  =lim f(x) ±lim g(x)
x a
x a
x a
Example
lim (3x + x) =
2
x  -1
9
5) lim [f(x)  g(x)] =lim f(x)  lim g(x)
xa
x a
x a
Example
lim [x (x-1)] =
4
x 2
10
f(x)
f(x) lim
6) lim
= x a
, where (lim g(x)  0)
x a g(x)
x a
lim g(x)
x a
Example
x +2
lim
=
x 3 x-2
11
7) lim cf(x) = c lim f(x) ; c constant
xa
xa
Example
lim 3(x +2x +1) =
2
x 4
12
8) lim n f(x) = n lim f(x) ,
x a
x a
n is positive integer and lim f(x)  0
x a
Example
lim
x 0
x+4 =
Attempt Q 1 AND Q2 (Tutorial)
13
Limits of Rational Function
Example
x 2 + 2 x + 4 (2)2 +2(2) + 4
l im
=
x2
2 +2
x+2
12
=
4
=
Note:
Limit of a rational function can be found
by substitution ONLY when the denominator is
not zero !!!
14
If f(x) and g(x) are polynomials, and c is any
number, then
lim
x c
f (x ) f (c )
=
, p r o v id e d
g (x ) g (c )
g (c )  0
But by using substitution, this is not always true!
Example:
x - 1 6  4  - 16
lim
=
=
x 4
4-4
x-4
2
2
indeterminate form
15
For cases where
lim f(x)
0
=
lim g(x) 0
x a
x a
Two methods can be used:
(a) Factorisation Method
(b) Multiplication Of Conjugates Method
(For function related to surd form)
16
Factorisation Method
Example 1
x -16
a) lim
x4
x-4
2
x +3x
b) lim 2
x -3 x +2x-3
2
x2 + 4x -12
c ) lim
x2
x2 -2x
17
Solution
x2 -16
a) lim
x4
x-4
Solve by substitution first
18
By using factorisation method:
x2 -16
lim
x 4
x-4
19
Try to substitute first!!!
x2 +3x
b) lim 2
x -3 x +2x-3
factorisation method:
20
Try to substitute first!!!
x + 4x -12
c ) lim
2
x2
x -2x
2
factorisation method:
21
Multiplication of Conjugates Method
(For function related to surd form)
Example 2
(a )
F in d
lim
x 4
x-4
x -2
(b ) F in d lim
x-2
x +2 -2
(c ) F in d
x+9-3
x
x2
lim
x 0
22
Solution
( a ) lim
x 4
x-4
x -2
Solve by substitution first
23
Multiplication of conjugates method:
x-4
lim
x 4
x -2
24
Try to substitute first!!!
( b ) l im
x2
x-2
x +2 -2
Multiplication of conjugates method:
25
Try to substitute first!!!
( c ) lim
x 0
x+9-3
x
Multiplication of conjugates method:
Attempt Q 3 AND Q4 (Tutorial)
26
8.1 Limits
LESSON 27
Learning outcomes
At the end of this topic, students should be able to :
1
Find one-sided limits
2
Determine the existence of the limit of a
function.
3
Find infinite limits
For more details, please visit
https://www.mathsisfun.com/calculus/limits.html
1
One-Sided Limits
A one sided limit can either be a right-hand limit
or left-hand limit
y
3
x
2- 2
lim- f(x) =
x →2
2+
lim+ f(x) =
x →2
2
Right-Hand Limit: (RHL)
The limit of f(x) as x approaches a from the
right side of a is equal to L
lim+ f(x) = L
x →a
Left-Hand Limit: (LHL)
The limit of f(x) as x approaches a from the
left side of a is equal to M
lim- f(x) = M
x →a
3
Refer the graph:
y
L
M
0 a-
lim- f(x) =
x→a
a
a+
x
lim+ f(x) =
x→a
4
Example 1 Find the value of the limits below:
x -5
(a ) limx →5 x -5
5
x - x-2
(b) lim+
x →2
x-2
2
6
Existence of a Limits
If
lim - f (x ) = lim + f ( x ) = L
x→ a
x→ a
then
l i m f ( x ) = L where a and L is a real number
x→ a
And therefore
is exist !!!
8
Example 2
The graph represents the function f(x).
Determine the existence of the limit.
y
a) lim f(x)
x →-1
3
b) lim f(x)
2
x→0
c) lim f(x)
1
-1
x→1
1
2
3
x d) lim f(x)
x→2
8
Solution
y
From the graph
2
a) lim f(x)
1
x →-1
x
-1
9
y
From the graph
b) lim f(x)
3
2
1
x→0
-1
0
x
1
10
From the graph
c) lim f(x)
x→1
y
3
2
1
x
0 1
2
11
From the graph
d) lim f(x)
x→2
y
3
2
1
0 1
2
x
3
12
Example 3
 x2 , if x  2
Given f(x) = 
, a =2
 4 , if x > 2
Determine:
i) lim- f(x)
x →a
ii) lim+ f(x)
x →a
iii) lim f(x)
x →a
13
Infinite Limits
Consider this example:
find lim
x→0
 1 
 2
x 
We must take left and right hand limits
as x approaches 0.
The following table presents some selected
values to help us to see the changing of f(x)
as x gets closer to 0 from left & right hand sides.
14
Graphically the function f(x) appears as below
 1 
f (x ) = lim  2 
x→ 0  x 
X -100 -10 1
f(x) 0.0001 0.01 1
If values of x continues
to take bigger values,
f(x) will continue to
increase without bound
y
X 1 10
100
f(x) 1 0.01 0.0001
If values of x continues
to take smaller values,
f(x) will continue to
increase without bound
-100 -10 -1 1 10 100
x
15
Definition of Infinite Limits
When the limit of f(x) increases to infinity as x approaches
a or the limit of g(x) decreases to negative infinity as x
approaches a then it is called Infinite Limits
16
y
lim - f (x ) = + 
lim + f (x ) = + 
x→ a
x→ a
x
a
lim f (x ) = + 
Remark:
x→ a
(i) If lim− f(x) =+ and lim+ f(x) =+
x →a
then lim f(x) =+
x →a
x →a
17
y
a
lim - f (x ) = - 
x→ a
lim f (x ) = - 
x
lim + f (x ) = - 
x→ a
x→ a
Remark:
(ii) If lim− f(x) =- and lim+ f(x) =-
x →a
then lim f(x) =-
x →a
x →a
18
y
or
a
x
a
Remark:
(iii) lim− f(x) =- and lim+ f(x) =+
x →a
x →a
OR lim− f(x) =+ and lim+ f(x) =-
x →a
lim− f(x)  lim+ f(x)
x →a
x →a
x →a
then lim f(x) does not exist
x →a
19
Example 4
Given f(x) =
1
.
x -1
Find lim + f ( x ) , lim - f ( x ) and lim f(x)
x→1
x→1
x →1
Solution
20
Example 5
1
. Find lim
Given f(x) =
x →2
x-2
1
x -2
Solution
21
Theorem 1
If lim f(x) =  and lim g(x) = L
x →a
x →a
then
lim[f(x) + g(x)] =lim f(x) +lim g(x)
x →a
x →a
= + L
x →a
=
22
Theorem 2
If lim f(x) =  and
x →a
lim g(x) = L , L  0
x →a
then
For L > 0
a) lim[f(x)g(x)]
x →a
=lim f(x) lim g(x)
x →a
=  .(L)
x →a
= +
For L < 0
b)lim[f(x)g(x)]
x →a
=lim f(x) lim g(x)
x →a
=  .(L)
=-
x →a
23
Example 6
2 
 1
+

 x −1 x +1 
Find lim+ 
x →1
Solution
Attempt Q 5 TO Q13 (Tutorial)
24
8.1 Limits: Asymptotes
Learning
outcomes
LESSON 28
At the end of this topic, students should be able to :
1
2
Find limits at infinity
f ( x)
f(x) and lim g(x)
Find lim
whenxlim
→+ 
x →+ 
x →+  g ( x )
are undefined.
3
Discuss the following limits :
 1 
lim  n  = 0 for n  0
x →−  x 
and
 1 
lim  n  = 0 and
x →+   x 
8.2 Asymptotes
4
Find the vertical and horizontal asymptotes
5
Sketch the graph of rational function involving
the vertical and/or horizontal asymptotes.
1
Limits at infinity
• The symbol  (infinity) does not represent a real
number.
• It’s a notation we use to denote how certain functions
behave.
“the limit of f as x approaches infinity”
= the limit of f as x moves increasingly far to the right
on the number line.
+
2
“the limit of f as x approaches negative infinity”
= the limit of f as x moves increasingly
far to the left on the number line.
-
How to find
lim f(x) and
x →+
lim f(x)
x →-
3
Consider the graph below:
y
What is the limit of f(x) when x
approaches +∞ ???
L
+
x
If the function f approaches L, when x
increases without limit, then
lim f(x) =
x →+
4
What is the limit of f(x) when x approaches -∞?
y
-
M
x
If the function f approaches M, when x
approaches -∞ , then
lim f(x) =
x →- 
5
Theorem For Limits At Infinity
for n > 0
 1 
i) lim  n  = 0
x →+  x 
1
ii) lim  n  = 0
x →-  x 
Example:
Example:
 1 
lim  5  =
x →+  x 
1
lim  5  =
x →-  x 
=
=
6
Example 1
5
Given f(x) =1.
x-1
Find lim f(x) and lim f(x).
x →+
x →-
Solution
5 

lim 1 −
=

x →+ 
x −1 
5 

lim 1 −
=

x →− 
x −1 
7
lim
x →+ 
f ( x)
f(x) and lim g(x) are undefined
when xlim
→
+

x →+ 
g ( x)
TYPE 1 If h(x) is a rational function,
f(x) 
lim
=
x → g(x)

- we divide the numerator and denominator by the
highest power of x in the denominator.
Remember !!!
x if x → +
x = x =
-x if x → -
2
8
Example 2
3x +1
.
x →+ 2x-5
Find lim
Solution
 3x + 1 
lim 
=

x →+ 2 x − 5


9
Example 3
 x 
Find lim 
.

2
x →-  3x -1 
Solution
 x 
lim  2  =
x →− 3 x − 1


10
Remember !!!
Example 4
 x +16 
lim 
.

x →- 
 x-13 
2
Find
 x if x → +
x = x =
-x if x → -
2
Solution
 x2 +16 
lim 
=

x →-  
x
-3


11
Example 5
Find
lim
x →+
4x +1
.
9-3x
Solution
4x +1
lim
=
x →+ 9-3x
12
TYPE 2
If h(x) is not a rational function
(i) Use direct substitution
(ii) If direct substitution fail,
NOTE:
Direct substitution fail if there is an
indeterminate case i.e.
(i)  - 
0
(ii)
0
(iii)  +  = 
use
(a) factorisation method
(b) multiplication of conjugates
13
Example 6
Find xlim
→+
(
)
x2 + 3 - x .
Solution
lim
x →+
lim
x →+
(
(
x2 + 3 - x
)
=
INDETERMINATE
)
x2 + 3 - x =
Attempt Q 14 (Tutorial)
14
Vertical and horizontal asymptotes
Types of Asymptotes:
1. Vertical Asymptotes
2. Horizontal Asymptotes
Horizontal Asymptotes, y=a
Vertical Asymptotes, x=b
15
Vertical Asymptotes
The line x = a is a vertical asymptotes for the graph of the function f(x) if
lim+ f(x) = +
x →a
OR
lim− f(x) = +
x →a
 f(x ) increase without bound as x → a+
or
x →a
-
lim+ f(x) = −
x →a
or
lim− f(x) = −
x →a
 f(x ) decrease without bound as x → a+
or
x →a
-
16
Vertical Asymptote for a Rational Function
STEPS:
(1) Make sure that the function is in the simplified form
Examples
( x 1)
(i) f ( x )
, x 1, 2
( x 1 )( x 2 )
(ii) f ( x )
1
x 2
, x 1, 2
(not in simplified
form)
(simplified form)
17
(2) Find the suspect of vertical symptote.
*(suspect = denominator)
denominator = 0
(3) Find domain of f(x). Next, examine
whether the domain approaching the suspect.
(4) If step 3 is satisfied, do infinite limits.
*(either LHL, RHL or both – depends on
the domain)
(5) V.A
x = suspect
18
Example 7
Determine the vertical asymptotes of f ( x ) =
4
x 2- 2 x
Solution
19
20
Example 8
Determine the vertical asymptotes of f ( x ) =
7x
x − 11
.
Solution
21
22
Horizontal Asymptotes
A line y = b is a horizontal asymptote of a function
y = f(x) if either
lim f ( x ) =b
x →+
OR
lim f ( x ) =b
x →-
NOTE!!
For horizontal asymptote, we are interested in the
behaviour as x → .
(limits at infinity)
23
Horizontal Asymptote for Rational Function
STEPS:
1. Find limits at infinity.
( lim f ( x ) and lim f ( x ))
x →+
x →−
a) Determine the highest power of x in the
denominator.
b) Divide the numerator and denominator of a
rational function by the highest power of x
occurs in the denominator.
2.
H.A
y=b
24
Example 9
3x
Determine the horizontal asymptote for f ( x ) =
x-2
Solution
Attempt Q 15 (Tutorial)
25
Sketch graph of rational function involving
the vertical and/or horizontal asymptote
STEPS:
1. Identify the domain of the function.
2. Find the infinite limits and identify the vertical
and horizontal asymptotes.
3. Identify the y-intercept and x-intercept of the
function.
4. Sketch the infinite limits obtained in step 2, then,
connect the sketches passing through the
y-intercept and and x-intercept to give the graph.
Example 9
Find the vertical and horizontal asymptotes (if exist) for f ( x ) =
x
.
3x − 2
Hence, sketch the graph.
Solution
27
Step 4: Sketch
Example 9
Find the vertical and horizontal asymptotes (if exist) for f ( x ) =
1
.
2
x −1
Hence, sketch the graph.
Solution
32
Graph:
Step 1:
Step 2: Vertical asymptote:
Horizontal asymptote:
Step 3: x-intercept :
y-intercept :
Step 4: Sketch
Attempt Q 16 (Tutorial)
8.3 Continuity
LESSON 29
Learning outcomes
At the end of this topic, students should be able to :
1
Interprete the continuity of a function at a point
2
Compute the continuity of a function at a point.
1
Definition
A function is said to be continuous if its graph can be
drawn over each interval of its domain.
Remarks:
Continuous motion of the pen without lifting the pen.
2
Which of these functions
are continuous?
a)
f(x)
b)
f(x)
x
x
3
c)
d)
f(x)
f(x)
x
x
4
Continuity at a point
Let f(x) be a function defined on an open interval
containing a.
The function is continuous at x = a if and only if :
i)
ii )
f (a) is defined
lim f ( x) exist
x→a
lim+ f ( x ) = lim - f ( x )
x→ a
x→ a
iii ) lim f ( x) = f (a)
x→a
For more details, please visit
https://www.analyzemath.com/calculus/continuity/
continuous_functions.html
5
Example 1
Determine whether the function f(x) is continuous
at x = -4, x = -1 and x = 2.
f(x)
4
3
-4
-2
-1
0
x
4
2
6
Solution
f(x)
at x=-4,
4
3
-4
-2
x
-1
7
at x=-1,
f(x)
4
-2
-1
3
0
2 x
8
at x=2,
2
9
Example 2
3 if x = 4
Sketch the graph of f ( x) = 
 x + 2 if x  4
Determine whether f(x) is continuous at x = 4.
Solution
10
11
Example 3
 x 2 +1,
Let f(x)= 
 x+3,
x 2
x 2
Proof that f is continuous at x=2.
Solution
12
13
Example 4
Determine the value of c if function
 x 2 -1 ,x  3
f(x)= 
2cx ,x>3
is continuous at x = 3.
Solution
14
Example 5
Determine the constants c and d such that the function
h(x) is continuous on the entire real line.
x4
 2 x,

h ( x ) = c x + d , 4  x  9
 x + 1,
x

9

Solution
15
Attempt Q 17 TO Q24 (Tutorial)
16