Advanced Mathematics D
Chapter Two
Limits & Continuity
Enter Calculus World
We have known “Function”
This is the tool for us enter the Calculus
World
The first adventure is “Limit”
Limits is the soul of Calculus as well as
“Advanced Mathematics”!
Ancient Limit Thought
 Zhuangzi （庄子,369-286 B.C）
in《天下》said:
一尺之锤，日取其半，万世不竭。
 Chinese ancient mathematician
Liu Hui （刘徽） in 263 a.c. said
in《九章算术》 :
“割之弥细，所失弥小. 割之又割，以
至于不可割，则与圆周合体，而无所
失矣”。
Calculation of a Circle Area
Inside
Polygo
n Area
Outsid
e
Polygo
n Area
Error to Circle
N=4
2
4
1.142/0.858
N=6
2.598
3.464
0.544/0.323
N=8
2.828
3.314
0.313/0.172
N=10
2.939
3.249
0.203/0.108
N=12
3
3.215
0.142/0.074
Zeno of Elea (490–430 BC)
paradoxes Achilles and the Tortoise ：
A is in a footrace with a slow T by 10* its speed.
A allows T a head start of 100 meters.
After A run 100 m, to T start point, meanwhile, T run 10 m.
A then run 10 m, and T has advanced farther;
A need more time to reach 3rd point, while T moves ahead.
 Thus, whenever A reaches somewhere T has been, he still
has farther to go. Therefore, as there are an infinite number
of points A must reach where T has already been, he never
overtake T.
Some Examples for Limit
Value of a function
Area Problem
Given a function f, find the area between the
graph of f and an interval on the x-axis
Tangent Line Problem
Give a function f, and a point on its graph, find
an equation of the line that is tangent to the
graph at the point
Tangent Line
Limit (An Informal View)
If the value of f (x) can be made as close
as we like to L by taking values of x
sufficiently to a (not equal to a), then we
write
lim f ( x)  L
x a
or
f ( x)  L, as
xa
Read as:
“the limit of f (x) as x approaches a is L.”
Equivalently
lim  f ( x)  L   0
x a
Examples of Limit Value
 as x  1,
x  1 ( x  1)( x  1)
f ( x) 

?
x 1
x 1
x
0.999
0.9999
0.99999
1 1.00001
1.0001
1.001
f (x)
1.999500
1.999950
1.999995
2 2.000005
2.000050
2.000500

as x  0,
sin x
g ( x) 
?
x
x
-0.001
-0.0001
-0.00001
0 0.00001
0.0001
0.001
g (x)
0.9999998
0.9999999
98
0.999999
99998
1 0.999999
99998
0.9999999
98
0.9999998
Example of No Limit Value
as x  0,


 
f ( x)  sin    ?
x
x
π/x
f(x)
±2
±π/2
±1
±π
±1
0
±2/3
±π/2
±10/100
±10π
±10/105
±(10+1/2)π
±10/1000
±100π
±10/1005
±(100+1/2)π
±10/10000
±1000π
±10/10005
±(1000+1/2)π
±10/100000
±10000π
±1
0
±1
0
±1
0
±1
0
Example of No Limit Value as x  0,
 
f ( x)  sin    ?
x
Example of “Part” Limit
 f1 ( x), if x  a,
f ( x)  
 f 2 ( x), if x  a.
lim f ( x)  f1 (a),
x a
lim f ( x)  f 2 (a).
x a
f 2 ( x)
f1 (a)  f 2 (a)
f1 ( x)
a
One Side Limit (An Informal View)
If the values of f (x) can be made as close
as we like to L by taking values of x
sufficiently close to a (but greater than a),
written as
lim f ( x)  L
xa
 If the values of f (x) can be made as close
as we like to L by taking values of x
sufficiently close to a (but less than a),
written as
lim f ( x)  L
xa
Example unequal 2-sided limit
| x|
lim
not exists,
x 0 x
| x|
x
| x|
x
lim
 lim  1  lim
 lim
 1
x 0 x
x 0 x
x 0
x 0
x
x
1
-1
Infinite Limit (An Informal View)
If the values of f (x) increases without
bound as x approaches a from left or right
or both, written as respectively
lim f ( x)  ,
x a
lim f ( x)  , lim f ( x)  
x a
x a
If the values of f (x) decreases without
bound as x approaches a from left or right
or both, written as respectively
lim f ( x)  ,
x a
lim f ( x)  , lim f ( x)  
x a
x a
Infinite Limit

1
lim  ,
x 0 x
0
x
f(x)
1
lim  
x 0 x
-1
-0.01
-0.0001
0 0.0001
0.01
1
1
-1000
-10000
∞
100
1
10000
The Relationship Between
1-sided and 2-Sided Limit
The 2-sided limit of a function f (x) exists at
a if and only if both of the one-sided limits
exist at a and have the same values , i.e.
lim f ( x)  L 
x a
lim f ( x)  L  lim f ( x)
x a
x a
Exercises & Questions
Suppose that a function f has the property
that for all real number x, the distance
between f (x) and 3 is at most |x|, we can
conclude that
as x  ? f ( x)  ?
The slope of the secant line through P(2,4)
and Q(x,x2) on y=x2 is x+2. It follows that
the slope of the tangent line to this
parabolic at point P is _____
Computing Limit
Simple facts
 if a and k are real numbers
lim k  k ,
lim x  a,
lim 1/ x  ,
lim 1/ x  .
x a
x 0
x a
x 0
Limit Properties
f ( x)  L1 , lim g ( x)  L2
 If lim
then
x a
x a

lim  f ( x)  g ( x)   lim f ( x)  lim g ( x)  L1  L2
xa
x a
x a

 lim g ( x)   L L
lim  f ( x) / g ( x)    lim f ( x)  /  lim g ( x)   L / L
lim  f ( x) g ( x)   lim f ( x)
xa
xa
xa
xa
lim n f ( x)  n lim f ( x)  n L1
xa
xa
x a
x a
1
2
1
2
( L2  0)
( L1  0, if n is even)
Limit of Polynomial Function
 Let
p( x)  c0  c1 x 
lim p( x)  c0  c1a 
x a
 cn x n , then
 cn a  p(a)
n
Limit of Rational Function
There are 3 cases for limit of rational
p( x) / q( x)
functions: lim
xa
Suppose that both lim p(x) and lim q(x) exist
Case 1: lim p(x)≠0,lim q(x) ≠0
lim [p(x)/q(x)]=lim p(x) /lim q(x)
Case 2: lim p(x) ≠0,lim q(x) =0
lim [p(x) / q(x)]=∞
Case 3: lim p(x) =0,lim q(x) =0
not determined
Limits Involving Radicals
Can be regard the radical term as a new
variable, then turn the function into a new
function of a polynomial function.
Limit of Piecewise-Defined Function
Identity the joint points of the piecewisedefined function
If limit value is for the joint point, find 2sided limit of the joint point
Notice that the limit of the joint value may
be not equal to the point value where the
function defined.
Limit at Infinity
If the values of f (x) eventually get as close
as we like to number L as x increases
without bound, then
lim f ( x)  L,
x 
or
f ( x)  L as
x  
Infinite Limits at Infinity
(An Informal View)
 If values of f (x) increase without bound as x
goes to infinite, then we write
lim f ( x)  
x 
 Similarly if f (x) decrease without bound as x
goes to infinite, we write
lim f ( x)  
x 
Limit of Polynomials as x→∞

lim x n  
x 
, if n is odd,
lim x  
x 
, if n is even.
n
 The end behavior of a polynomial matches
the end behavior of its highest degree term:
lim (c0  c1 x 
x
 cn x )  lim cn x
n
x 
n
Limits of Rational Function as x→∞
One technique for determining the end
behavior of a rational function is to divide
each term in the numeration and
denominator by the highest power of x that
occurs in the denominator, after which the
limiting behavior can be determined using
results we have already established.
Becareful on ∞±∞
 Like limitation of 0/0, end behavior of ∞±∞ is
not determined
 Ex.
n
lim ( x  x)  
x 
lim ( x  x)  0
x 
 x 1 
lim 
 x 1
x 
 x 1

2
Limits Involving Radicals
It would be helpful to manipulate the
function so that the powers of x are
transformed to powers of 1/x
End Behaviors of Trigonometric,
Exponential and Logarithmic Fun.
 No limit for
lim sin x
x 


lim e  ,
x
x 
lim ln x  ,
x 
lim e  0
x
x 
lim ln x  
x 0
Definition of Continuity
A function is said to be continuous at x=c
provided the following conditions are
satisfied
 f (c) is defined

lim f ( x) exists
x c

lim f ( x)  f (c)
x c
Continuity in an interval
A function f is said to be continuous on a
closed interval (a,b), if for any point c ∈(a,b),
 f is continuous on c
Continuity on an Interval
A function f is said to be continuous on a
closed interval [a,b], if the following
conditions are satisfied:
 f is continuous on (a,b)
 f is continuous from the right at a
 f is continuous from the left at b
Continuity on other Intervals
In the same way, the continuity of f can be
extend
(a,b], [a,b),
(-∞,b], [a, +∞), (-∞, +∞),
Properties of Continuity
 If function f and g are continuous at c,
then
 f±g is countinuous at c
 fg is continuous at c
 f/g is continuous at c if g(c)≠0
 f/g has a discontinuity at c if g(c)=0
Continuity of Polynomials & Rational
Functions
 A polynomial is continuous everywhere
 A rational function is continuous at every
point where the denominator is nonzero,
and has discontinuities at the point where
the denominator is zero.
Continuity of Compositions
g ( x)  L
 If lim
x c
at L, then
and if the function is continuous

lim f  g ( x)   f lim g ( x)
x c
x c
 Same for sided limits and infinite limits

Continuity of Compositions (cont.)
If function g is continuous at c and function
f is continuous at g(c), then f◦g is
continuous at c
If f and g continuous everywhere then f◦g
is continuous everywhere
Intermediate-Value Theorem
 If f is continuous on a closed interval [a,b]
and k is any number between f(a) and f (b),
inclusive, then there is at least one
number x in [a,b] such that f (x)=k.
Intermediate-Value Theorem (cont.)
If f is continuous on [a,b] and if f (a) and
f (b) are nonzero and have opposite signs,
then there is at least one solution of the
equation f (x)=0 in the interval (a,b)
Continuity of Trigonometric Function
 IIf c is any number in the natural domain of the
stated trigonometric function, then
limsin x  sin c, lim cos x  cos c, lim tan x  tan c,
x c
x c
x c
lim csc x  csc c, limsec x  sec c, lim cot x  cot c.
x c
x c
x c
Continuity of Inverse Function
If f is one-to-one function that is
continuous at each point of its domain,
then f -1 is continuous at each point of its
domain, that is f -1 is continuous at each
point in the range of f
Sequeezing Theorem
 Let f,g and h be functions satisfying
g ( x )  f ( x )  h( x )
for all x in some open interval containing the
munber c, with the possible exception that the
inequalities need not hold at c. If g and h have
the same limit as x approaches c, say
lim g ( x)  lim h( x)  L,
x c
x c
then f also has this limit as x approaches c
lim f ( x)  L.
x c
Example
sin x
1  cos x
lim
 1, lim
0
x 0
x 0
x
x