MATH 122: CALCULUS I
Limits & Continuity
Lilian F. Kyei (Ms)
Second Semester, 2025/2026
Department of Mathematics
University of Ghana
Recap
Determine the following limits by sketching the graphs of the
functions.
tan x
15. lim
x!0 x
Use
cos x 1
16. lim
x!0
x2
to
sin(2x)
17. lim
x!0 tan(4x)
Geogebra
visualise
functor
These
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Theorem
Suppose f (x)  g(x) for all x in an open interval containing a,
except possibly at a. If lim f (x) and lim g(x) exists, then
x!a
x!a
lim f (x)  lim g(x)
x!a
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x!a
36
Theorem (The Squeeze Theorem)
Suppose f (x)  g(x)  h(x) for all x in an open interval
containing a, except possibly at a, and
lim f (x) = lim h(x) = L.
x!a
x!a
Then
lim g(x) = L
x!a
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S
B
Theorem
joa
sin ✓
lim
=1
✓!0 ✓
We note
that
Area
o
xOAB
ziesin a
=>
=>
Sinc
I
Sector OAB
OAB
/Sino
& triangle
Ot
Area
triangle
Smo
Area
of
=
Areay Sesto OAB)
Nu ,
tim
1 =
O < temp
soto
So
<
to
=
=
PtO
=>
to
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Area
1
Area
of
DOAS
andhim=
D
him
=
a
1Bytherem
38
Cos2X
=
-
Theorem
=
cos ✓
✓!0
✓
lim
1
=
O
Go =in
=
on
a
As00 ,
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CisX- Sin'x
CrsO
1-Sinx-Sinx
1
=
=
=
2Sinx
Cos(2())
1-2 Sin &
him Glm-Si a
=him So limin
a
=
=
-
-
8
8
oh 50
39
famustexte
f
*himfs
Let f be a function defined on an open interval containing all
values of x close to a. f is continuous at a if
Definition (Definition of continuity at a point)
=
lim f (x) = f (a)
x!a
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40
↳ Usually
Removable
feal + limf(x)
/weldm
define
Type of discontinuities
X99
entinie
Jump discontinuity
↳
canwhe lim
Urremovable
t
m
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anounfinite discontinuit
41
Activity Sheet 3 - Concept of Continuity
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42
fa
=
\ If * A
Function
a
4(x2 4)
f (x) =
x 2
f (x) =
(
f(x)
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if x = 2
G4x +
lim f (x)
x!a
f (a)
Remark
fined Discontinuous
16 16
[Removable]
Discontinuous
16
2
8
=
lim f (x)
x!a
16
2
4x + 8 if x 6= 2
4
lim f (x)
x!a+
16 16
4 [Removable
if X # 2
if
x=
2
43
Function
f (x) =
f (x) =
(
0
if x < 0
1
if x
8
>
>
<3
=
0
0
if x = 1
f (a)
Remark
1
O hmisnt
Discontinuous
2
22/
lim f (x)
x!a+
if x < 1
x
1
>
>
: 2 + px
fx
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a
1
1 if x > 1
4I
lim f (x)
x!a
lim f (x)
x!a
[Jump
discontinuite]
Discontinuous
[Removable
discontinuity]
if ncl
44
Function
a
x 1
f (x) = 2
x +x 2
1
x 1
f (x) = 2
x +x 2
-2
!
f (x) =
f(x
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p
=
x+2 2
x 2
2
lim f (x)
x!a+
lim f (x)
x!a
"It
G
f (a)
Remark
Discontinuous
def
i
n
ed
[removable
"3
discontinuityT
"
*
lim f (x)
x!a
a
-
-
-
fiared Discontinuous
dimtot [Infinite
discontinuity]
fined Discontinuous
"4 [Removable
discontinuity]
de
45
Function
f (x) =
p
a
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3
x
x
x3 + x2
f (x) =
|x
f (x) =
p
3+x
x2
x+2
3x + 1
1|
0
1
1
lim f (x)
x!a+
· is
lim f (x)
x!a
lim f (x)
x!a
"es les
2
.
f (a)
fied Discontinuous
de
un
chimnot
it unde
111
Remark
Removable
L
discontinuite]
Discontinuous
fined [Jump
I
discontinuite]
Continuous
46
Function
f (x) =
f (x) =
p
a
p
3+x
3
x
x
(x + 1)2
1
3x
x2 + 1
a
lim f (x)
x!a
f (a)
Remark
The
&
in
f (x) =
p
x2 + 2x
p
x2
2x
2
f (x) =
p
x2 + 2x
p
x2
2x
2
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lim f (x)
x!a
0
x
f (x) = p
lim f (x)
x!a+
Fill gaps You
may use
Geogebra fur
Visualisation
47
Definition (Continuity from the Right and from the Left)
A function f is continuous from the right at a if
lim f (x) = f (a).
x!a+
A function f is continuous from the left at a if
lim f (x) = f (a).
x!a
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Definition (Continuity on an interval)
A. A function f is continuous on an open interval (a, b) if it is
continuous at every number in the interval.
B. A function f is continuous on an closed interval [a, b] if it is
continuous on (a, b) and also continuous from the right at a
and from the left at b.
C. A function f is continuous on an a half-open interval [a, b)
or (a, b] if it is continuous on (a, b) and continuous from the
right at a and continuous from the left at b respectively.
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Theorem
If f and g are continuous at a, then the following functions are
also continuous at a
a. f ± g
b. f g
c. cf , where c is a constant
f
d. , if g(a) 6= 0
g
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Theorem (Continuity of Polynomial and Rational Functions)
A. A polynomial function is continuous on ( 1, 1)
B. A rational function is continuous on its domain.
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Theorem (Continuity of Trigonometric Functions)
The functions sin x, cos x, tan x, sec x, csc x, and cot x are
continuous at every number in their respective domain.
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Theorem (Limit of a Composite Functions)
If the function f is continuous at L and lim g(x) = L, then
x!a
lim f (g(x)) = f (L)
x!a
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Theorem (Continuity of Composite Functions)
If the function g is continuous at a and the function f is
continuous at g(a), then the composition f g is continuous at
a.
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