Chabot Mathematics
§1.6 Limits
& Continuity
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Review §
1.5
 Any QUESTIONS About
• §1.5 → Limits
 Any QUESTIONS About
HomeWork
• §1.5 → HW-05
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
§1.6 Learning Goals
 Compute and use
one-sided limits
 Explore the concept of
continuity and examine
the continuity of several
functions
 Investigate the
intermediate
value property
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Limits
 Limits are a very basic aspect of
calculus which needs to be taught first,
after reviewing old material.
 The concept of limits is very important,
since we will need to use limits to make
new ideas and formulas in calculus.
 In order to understand calculus, limits
are very fundamental to know!
Chabot College Mathematics
4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Continuous Functions
 Generally Speaking A function is very
likely to be “continuous” if:
The graph has no holes
or gaps and can be
drawn on a piece of
paper without lifting
The Drawing Instrument
(Pencil or Pen)
Chabot College Mathematics
5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Smooth Functions
 Generally Speaking A function is very
likely to be “smooth” if:
The graph of the function is a “flowing”
curve. This means that the graph of the
function does not contain any “sharp”
corners
• Smoothness Analysis will
be covered after we learn
how to evaluate the
“Slope” of curved lines
Chabot College Mathematics
6
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Continuous vs. DisContinuous
 CONTINUOUS
Function Plot
Chabot College Mathematics
7
 DIScontinuous
Function Plot
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Smooth vs. Kinked/Cornered
 SMOOTH-Curved
Function Plot
Chabot College Mathematics
8
 SHARP-Cornered
Function Plot
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
ONEsided Limits - From LEFT
 If f(x) Approaches L
as x→c from the
Left; i.e., x<c, write:
x c
• See Graph at Right
Chabot College Mathematics
9
3
y = f(x)
lim - f  x 
MTH15 • Bruce Mayer, PE • OneSided Limits
4
2
1
0
-1
-1
X: 1.5
Y: 1.034
X: 1.285
Y: 0.8547
X: 0.9539
Y: 0.6419
X: 0.6333
Y: 0.5223
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
1
2
3
x 1
lim  2
 1.034
x 1.5 x  2
3
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
4
ONEsided Limits – From RIGHT
 If f(x) Approaches L
as x→c from the
Left; i.e., x<c, write:
x c
• See Graph at Right
Chabot College Mathematics
10
3
X: 2.607
Y: 2.128
y = f(x)
lim f x 
MTH15 • Bruce Mayer, PE • OneSided Limits
4
2
X: 2.066
Y: 1.566
X: 2.337
Y: 1.844
X: 1.5
Y: 1.034
X: 1.766
Y: 1.271
1
0
-1
-1
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
1
2
x
3
x 1
lim  2
 1.034
x 1.5 x  2
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
4
Example  PieceWise Fcn
1  x 2 , if x  1
f ( x)  
3x  1 , if x  1
MTH15 • Bruce Mayer, PE • 2-Sided Limit
10
9
8
7
f(x)  PieceWise
 Find the OneSided
Limits for Function:
 Compute the
one-sided limits
of f(x) as x
approaches 1
6
5
4
3
2
1
0
-1
-2
-3
-4
-3
XY f cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
-2
-1
0
1
x
Chabot College Mathematics
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
2
3
Example  OneSided Limits
 SOLUTION
f  x  and lim f  x 
 Need to Determine: xlim
1
x 1
 Because the function is defined by the
first expression for values of x ≤1, have


lim f ( x)  lim (1  x )  1  (1)  0
2
x1
2
x1
 Also the fcn is defined by the second
expression for values of x >1, have
lim f ( x)  lim (3x  1)  3(1)  1  4
x 1
Chabot College Mathematics
12
x 1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Example  OneSided Limits
1  x 2 , if x  1
f x   
3x  1 , if x  1
 SOLUTION
lim f x   ??
x 1
 ReCall the
Requirement for Limit Existence
lim f x  
x 1
lim f x 
x 1
 For the Given Fcn use the Transitive
Property to Recognize that
the Limit x→1 Does Not Exist as
lim f x   0  4 
x 1
Chabot College Mathematics
13
lim f x 
x 1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Chabot College Mathematics
14
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 01Jul13
% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m
%
% The Limits
xmin = -3; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -4; ymax = 10;
% The FUNCTION
x1 = linspace(xmin,xmax1,500); y1 = 1-x1.^2;
x2 = linspace(xmin2,xmax,500); y2 = 3*x2+1;
% The Total Function by appending
x = [x1, x2]; y = [y1, y2];
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x1,y1,'b', x2,y2,'b', zxv,zyv, 'k', zxh,zyh, 'k', x1(end),y1(end),
'ob', 'MarkerSize', 12, 'MarkerFaceColor', 'b',...
'LineWidth', 3),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrow
PieceWise'),...
title(['\fontsize{14}MTH15 • Bruce Mayer, PE • 2-Sided Limit',]),...
annotation('textbox',[.51 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor',
'none', 'String',
'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7)
hold on
plot(x2(1),y2(1), 'ob', 'MarkerSize', 12, 'MarkerFaceColor', [0.8 1 1],
'LineWidth', 3)
set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:1:ymax])
hold off
Continuity Analysis
 DEFININITION: A function, f(x) is
continuous at a point c If and Only If The
limit of f(x) is independent of the direction
of Approach; that is the fcn is continuous if:
lim f  x   lim f  x 
x c
x c
• Note that this a Necessary AND Sufficient,
Condition
Chabot College Mathematics
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Example  Continuity
MTH15 • Bruce Mayer, PE • Continuity Analysis
1000
 Consider Function:
 Determine if the
Function is
Continuous at
• x=4
• x=5
Chabot College Mathematics
16
y = f(x) = (27x - 343)/(x - 5)
27 x  343
f x  
x 5
• See Graph at Right
800
600
400
200
0
-200
-400
-600
-800
-1000
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
1
2
3
4
5
6
7
8
x
 Use BiLateral
Approach Limit Test
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
9
10
Example  Continuity
 Find for x = 4 The
BiLateral Limits
27 x  343
lim
&
x 4
x 5
27 x  343
lim
x4
x5
 At x = 3.9999
273.9999   343
f x  
 234.979
3.9999  5
 At x = 4.0001
274.0001  343
f x  
 235.021
4.0001  5
Chabot College Mathematics
17
 By the PolyNomial
Limit Rule
27 x  343 274  343  235


 235
x4
x5
45
1
lim
 The Left Approach
(3.9999) and the
Right Approach
(4.0001) Both Lead
to 235, thus the fcn
IS Continuous at
x=4
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Example  Continuity
MTH15 • Bruce Mayer, PE • Continuity Analysis
1000
 Now Check
Continuity at x = 5
y = f(x) = (27x - 343)/(x - 5)
• Use Approach Tables
From LEFT
x
4
4.5
4.8
4.9
4.99
4.999
4.9999
f (x )
235
443
1067
2107
20827
208027
2080027
From RIGHT
x
5.0001
5.001
5.01
5.1
5.2
5.5
6
f x  
800
f (x )
600
400
200
0
-200
-400
-600
-800
-2079973
-207973
-20773
-2053
-1013
-389
-181
-1000
XY f cnGraph6x6BlueGreenBkGndTemplate1306.m
0
1
2
3
4
5
 From
lim
Approach x5
Tables
lim
x 5
Note:
18
6
7
8
9
10
x


Chabot College Mathematics
27 x  343
x 5
27 x  343
 
x 5
27 x  343
 
x 5
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
PieceWise Continuity
 A NONontinuous PieceWise-Defined
Function can be made continuous thru
the process of Break-Point Matching.
 BreakPoint Matching
• One Fcn Left Unchanged
• At Least ONE Variable-Term in the other
Fcn is multiplied by a CONSTANT
MTH15 • Bruce Mayer, PE • PcWise Continuous
Chabot College Mathematics
19
9
8
f(x)  PieceWise
• The two Fcns are
then equated at the
BreakPoint Value
10
7
6
5
4
3
2
1
0
-2
XY f cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
-1
0
1
x
2
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Example  Make Continuous
f P x   P  3x 2  5 x  11
f Q x  
Chabot College Mathematics
20

Q x 7

f x 
x 1
x 1
MTH15 • Bruce Mayer, PE • DIScontinuous
15
10
f(x)  PieceWise
 Consider the Fcn:
 This Fcn is
NONcontinuous as
shown in the Plot
 Make this Plot
Continuous for
Constants P & Q:
3x 2  5 x  11 if
 
x 7
if

5
0
-5
-10
-15
-3
XY f cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
-2
-1
0
1
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
2
3
Example  Continuous at 8
f x 
 The FineTuned Fcn
 The
Plot
24 x 2  5 x  11 if
 
x 7
if

MTH15 • Bruce Mayer, PE • PcWise Continuous
15
f(x)  PieceWise
10
5
0
-5
-10
-15
-2
Chabot College Mathematics
21
XY f cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
-1
0
1
x
2
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
x 1
x 1
Example  Continuous at −13
 The FineTuned Fcn
 The
Plot
 x 2  5 x  11
 
 13 8 x  7
f x 


MTH15 • Bruce Mayer, PE • PcWise Continuous
10
f(x)  PieceWise
5
0
-5
-10
-15
-20
-2
Chabot College Mathematics
22
XY f cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
-1
0
1
x
2
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
if
x 1
if
x 1
Chabot College Mathematics
23
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
P MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 01Jul13
% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m
%
% The Limits
xmin = -2; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -15; ymax = 15;
% The FUNCTION
x1 = linspace(xmin,xmax1,500); y1 = 24*x1.^2 - 5*x1 - 11 ;
x2 = linspace(xmin2,xmax,500); y2 = sqrt(x2) + 7;
% The Total Function by appending
x = [x1, x2]; y = [y1, y2];
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x1,y1,'b', x2,y2,'b', zxv,zyv, 'k', zxh,zyh, 'k',...
'LineWidth', 3),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrow
PieceWise'),...
title(['\fontsize{14}MTH15 • Bruce Mayer, PE • PcWise Continuous',]),...
annotation('textbox',[.41 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor',
'none', 'String',
'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7)
hold on
set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax])
hold off
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Q MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 01Jul13
% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m
%
% The Limits
xmin = -2; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -20; ymax = 10;
% The FUNCTION
x1 = linspace(xmin,xmax1,500); y1 = 3*x1.^2 - 5*x1 - 11 ;
x2 = linspace(xmin2,xmax,500); y2 = (-13/8)*(sqrt(x2) + 7);
% The Total Function by appending
x = [x1, x2]; y = [y1, y2];
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x1,y1,'b', x2,y2,'b', zxv,zyv, 'k', zxh,zyh, 'k',...
'LineWidth', 3),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrow
PieceWise'),...
title(['\fontsize{14}MTH15 • Bruce Mayer, PE • PcWise Continuous',]),...
annotation('textbox',[.41 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor',
'none', 'String',
'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7)
hold on
set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax])
hold off
Intermediate Value Theorem
 If f(x) is a continuous function on a closed
interval [a, b] and L is any number between
f(a) and f(b), then there is at least one
number c in [a, b] such that f(c) = L
y
y  f ( x)
f(b)
f(c) = L
f(a)
x
a
Chabot College Mathematics
25
c
b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Example  IVT
Given Fcn → f x   3x 2  2 x  5
Show That f(x)=0 has a solution on [1,2]
SOLUTION
Since the Function is a PolyNomial the
Fcn IS Continuous for all x
 Check Interval EndPoints




f 1  31  21  5  4  0
2
f 2  32  22  5  3  0
2
Chabot College Mathematics
26
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Example  IVT
MTH15 • Bruce Mayer, PE • IVT
15
10
y = f(x)=3x2 - 2x - 5
 STATE: f(x) is
continuous
(polynomial) and
since f(1) < 0 and
f(2) > 0, by the
Intermediate Value
Theorem there
exists c on [1, 2]
such that f(c) = 0.
5
0
(c,0)
-5
-10
XY f cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
0
1
2
x
Chabot College Mathematics
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
3
MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 01Jul13
% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m
%
% The Limits
xmin = 0; xmax1 = 3; xmin2 = xmax1; xmax = 3; ymin = -10; ymax = 15;
% The FUNCTION
x1 = linspace(xmin,xmax1,500); y1 = 3*x1.^2 - 2*x1 - 5 ;
x2 = linspace(xmin2,xmax,500); y2 = 3*x2.^2 - 2*x2 - 5;
% The Total Function by appending
x = [x1, x2]; y = [y1, y2];
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x1,y1,'b', x2,y2,'b',zxh,zyh, 'k',...
'LineWidth', 3),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}y = f(x)=3x^2 - 2x 5'),...
title(['\fontsize{14}MTH15 • Bruce Mayer, PE • IVT',]),...
annotation('textbox',[.41 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor',
'none', 'String',
'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7)
hold on
set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax])
hold off
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
WhiteBoard Work
 Problems From §1.6
• P13 → Find Limit Using Algebra
• P52 → Electrically Charged Sphere
• P56 → Create Continuity
Chabot College Mathematics
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
All Done for Today
Know
Your
Limits
Chabot College Mathematics
30
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
–
Chabot College Mathematics
31
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Make Continuous - P
Chabot College Mathematics
32
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Make Continuous - Q
Chabot College Mathematics
33
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Chabot College Mathematics
34
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Chabot College Mathematics
35
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Charge Hollow Sphere E-fld
MTH15 • Bruce Mayer, PE • P1.6-52 Charged Sphere
1.1
1
y = E(x) (Volt/meter)
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
XY f cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
0
1
2
3
Chabot College Mathematics
x/R
36
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Bruce Mayer, PE
MATLAB Code
% Bruce Mayer, PE
% MTH-15 • 01Jul13
% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m
%
clear; clc;
% InDep Var = x/R
% The Limits
xmin = 0; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -.1; ymax = 1.1;
% The FUNCTION
x1 = linspace(xmin,xmax1,500); y1 = 0*x1 ;
x2 = linspace(xmin2,xmax,500); y2 = 1./x2.^2;
x3 = 1; y3 = 1/(2*1^2)
% The Total Function by appending
x = [x1, x2]; y = [y1, y2];
%
% The ZERO Lines
zxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];
%
% the 6x6 Plot
axes; set(gca,'FontSize',12);
whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Green
plot(x1,y1,'b', x2,y2,'b',...
'LineWidth', 3),axis([xmin xmax ymin ymax]),...
grid, xlabel('\fontsize{14}x/R'), ylabel('\fontsize{14}y = E(x)
(Volt/meter)'),...
title(['\fontsize{14}MTH15 • Bruce Mayer, PE • P1.6-52 Charged Sphere',]),...
annotation('textbox',[.41 .05 .0 .1], 'FitBoxToText', 'on', 'EdgeColor', 'none',
'String', 'XYfcnGraphBlueGreenBkGndSolidMarkerTemplate1306.m','FontSize',7)
hold on
plot(x3,y3, 'ob', 'MarkerSize', 6, 'MarkerFaceColor', 'b', 'LineWidth', 3)
plot(x2(1),y2(1), 'ob', 'MarkerSize', 6, 'MarkerFaceColor', [0.8 1 1], 'LineWidth',
3)
plot(x1(end),y1(end), 'ob', 'MarkerSize', 6, 'MarkerFaceColor', [0.8 1 1],
'LineWidth', 3)
set(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:.1:ymax])
hold off
Chabot College Mathematics
37
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Chabot College Mathematics
38
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
P1.6-52(B)
Chabot College Mathematics
39
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
P1.6-56 Continuous Plot
MTH15 • Bruce Mayer, PE • P1.6-56 PcWise Continuity
10
f(x)  PieceWist
0
-10
-20
-30
-40
-50
XY f cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
0
Chabot College Mathematics
40
1
2
3
4
x
5
6
7
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx
Chabot College Mathematics
41
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx