[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx1
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
§1.6 Limits&
Continuity
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx2
Bruce Mayer, PE Chabot College Mathematics
Review §
Any QUESTIONS About• §1.5 → Limits
Any QUESTIONS About HomeWork• §1.5 → HW-05
1.5
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx3
Bruce Mayer, PE Chabot College Mathematics
§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
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx4
Bruce Mayer, PE Chabot College Mathematics
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!
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx5
Bruce Mayer, PE Chabot College Mathematics
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)
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx6
Bruce Mayer, PE Chabot College Mathematics
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
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx7
Bruce Mayer, PE Chabot College Mathematics
Continuous vs. DisContinuous
CONTINUOUS Function Plot
DIScontinuous Function Plot
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx8
Bruce Mayer, PE Chabot College Mathematics
Smooth vs. Kinked/Cornered
SMOOTH-Curved Function Plot
SHARP-Cornered Function Plot
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx9
Bruce Mayer, PE Chabot College Mathematics
ONEsided Limits - From LEFT If f(x) Approaches L
as x→c from the Left; i.e., x<c, write:
• See Graph at Right
xfx -clim
-1 0 1 2 3 4-1
0
1
2
3
4
X: 1.5Y: 1.034
x
y =
f(x)
MTH15 • Bruce Mayer, PE • OneSided Limits
X: 1.285Y: 0.8547
X: 0.9539Y: 0.6419X: 0.6333
Y: 0.5223
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
034.12
1lim
2
3
5.1
x
xx
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx10
Bruce Mayer, PE Chabot College Mathematics
ONEsided Limits – From RIGHT If f(x) Approaches L
as x→c from the Left; i.e., x<c, write:
• See Graph at Right
xfx clim
-1 0 1 2 3 4-1
0
1
2
3
4
X: 1.5Y: 1.034
x
y =
f(x)
MTH15 • Bruce Mayer, PE • OneSided Limits
X: 2.066Y: 1.566
X: 1.766Y: 1.271
X: 2.337Y: 1.844
X: 2.607Y: 2.128
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
034.12
1lim
2
3
5.1
x
xx
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx11
Bruce Mayer, PE Chabot College Mathematics
Example PieceWise Fcn
Find the OneSided Limits for Function:
Compute the one-sided limits of f(x) as x approaches 1
1 if , 13
1 if , 1)(
2
xx
xxxf
-3 -2 -1 0 1 2 3-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
x
f(x)
P
iece
Wis
e
MTH15 • Bruce Mayer, PE • 2-Sided Limit
XYf cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx12
Bruce Mayer, PE Chabot College Mathematics
Example OneSided Limits
SOLUTION Need to Determine: Because the function is defined by the
first expression for values of x ≤1, have
Also the fcn is defined by the second expression for values of x >1, have
xfxfxx 11lim and lim
0)1(1)1(lim)(lim 22
11
xxf
xx
41)1(3)13(lim)(lim11
xxf
xx
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx13
Bruce Mayer, PE Chabot College Mathematics
Example OneSided Limits
SOLUTION ReCall the
Requirement for Limit Existence
For the Given Fcn use the Transitive Property to Recognize that the Limit x→1 Does Not Exist as
??lim 1 if , 13
1 if , 11x
2
xfxx
xxxf
xfxfxx
11
lim40lim
xfxfxx
11
limlim
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx14
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-15 • 01Jul13% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m%% The Limitsxmin = -3; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -4; ymax = 10;% The FUNCTIONx1 = linspace(xmin,xmax1,500); y1 = 1-x1.^2;x2 = linspace(xmin2,xmax,500); y2 = 3*x2+1;% The Total Function by appendingx = [x1, x2]; y = [y1, y2];% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onplot(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
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx15
Bruce Mayer, PE Chabot College Mathematics
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:
• Note that this a Necessary AND Sufficient, Condition
xfxfxx
c c
limlim
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx16
Bruce Mayer, PE Chabot College Mathematics
Example Continuity Consider Function:
• See Graph at Right
Determine if the Function is Continuous at• x = 4• x = 5
Use BiLateral Approach Limit Test
5
34327
x
xxf
0 1 2 3 4 5 6 7 8 9 10-1000
-800
-600
-400
-200
0
200
400
600
800
1000
x
y =
f(x)
= (
27
x -
34
3)/
(x -
5)
MTH15 • Bruce Mayer, PE • Continuity Analysis
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx17
Bruce Mayer, PE Chabot College Mathematics
Example Continuity Find for x = 4 The
BiLateral Limits
At x = 3.9999
At x = 4.0001
By the PolyNomial Limit Rule
The Left Approach (3.9999) and the Right Approach (4.0001) Both Lead to 235, thus the fcn IS Continuous at x = 4
5
34327lim&
5
34327lim
44
x
x
x
xxx
234.979
59999.3
3439999.327
xf
235.021
50001.4
3430001.427
xf
235
1
235
54
343427
5
34327lim
4
x
xx
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx18
Bruce Mayer, PE Chabot College Mathematics
Example Continuity Now Check
Continuity at x = 5• Use Approach Tables
From Approach Tables Note:
0 1 2 3 4 5 6 7 8 9 10-1000
-800
-600
-400
-200
0
200
400
600
800
1000
x
y =
f(x)
= (
27
x -
34
3)/
(x -
5)
MTH15 • Bruce Mayer, PE • Continuity Analysis
XYf cnGraph6x6BlueGreenBkGndTemplate1306.m
5
34327
x
xxf
x f (x )4 235
4.5 4434.8 10674.9 2107
4.99 208274.999 208027
4.9999 2080027
From LEFT
x f (x )5.0001 -2079973
5.001 -2079735.01 -20773
5.1 -20535.2 -10135.5 -389
6 -181
From RIGHT
5
34327lim
5
34327lim
5
5
x
xx
x
x
x
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx19
Bruce Mayer, PE Chabot College Mathematics
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• The two Fcns are
then equated at the BreakPoint Value
-2 -1 0 1 2 30
1
2
3
4
5
6
7
8
9
10
x
f(x)
P
iece
Wis
eMTH15 • Bruce Mayer, PE • PcWise Continuous
XYf cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx20
Bruce Mayer, PE Chabot College Mathematics
Example Make Continuous
Consider the Fcn: This Fcn is
NONcontinuous asshown in the Plot
Make this Plot Continuous for Constants P & Q:
1if7
1if1153 2
xx
xxxxf
7
1153 2
xQxf
xxPxf
Q
P-3 -2 -1 0 1 2 3
-15
-10
-5
0
5
10
15
x
f(x)
P
iece
Wis
e
MTH15 • Bruce Mayer, PE • DIScontinuous
XYf cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx21
Bruce Mayer, PE Chabot College Mathematics
Example Continuous at 8 The FineTuned Fcn The
Plot
1if7
1if11524 2
xx
xxxxf
-2 -1 0 1 2 3-15
-10
-5
0
5
10
15
x
f(x)
P
iece
Wis
eMTH15 • Bruce Mayer, PE • PcWise Continuous
XYf cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx22
Bruce Mayer, PE Chabot College Mathematics
Example Continuous at −13 The FineTuned Fcn The
Plot
1if7813
1if1152
xx
xxxxf
-2 -1 0 1 2 3-20
-15
-10
-5
0
5
10
x
f(x)
P
iece
Wis
eMTH15 • Bruce Mayer, PE • PcWise Continuous
XYf cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx23
Bruce Mayer, PE Chabot College Mathematics
P M
AT
LA
B C
od
e% Bruce Mayer, PE% MTH-15 • 01Jul13% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m%% The Limitsxmin = -2; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -15; ymax = 15;% The FUNCTIONx1 = linspace(xmin,xmax1,500); y1 = 24*x1.^2 - 5*x1 - 11 ;x2 = linspace(xmin2,xmax,500); y2 = sqrt(x2) + 7;% The Total Function by appendingx = [x1, x2]; y = [y1, y2];% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onset(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax])hold off
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx24
Bruce Mayer, PE Chabot College Mathematics
Q M
AT
LA
B C
od
e% Bruce Mayer, PE% MTH-15 • 01Jul13% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m%% The Limitsxmin = -2; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -20; ymax = 10;% The FUNCTIONx1 = 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 appendingx = [x1, x2]; y = [y1, y2];% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onset(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax])hold off
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx25
Bruce Mayer, PE Chabot College Mathematics
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 f x
a b
f(a)
f(b)
c
f(c) = L
x
y
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx26
Bruce Mayer, PE Chabot College Mathematics
Example IVT
Given Fcn → 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
523 2 xxxf
03522232
045121312
2
f
f
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx27
Bruce Mayer, PE Chabot College Mathematics
Example IVT 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.
0 1 2 3-10
-5
0
5
10
15
x
y =
f(x)
=3
x2 - 2
x -
5
MTH15 • Bruce Mayer, PE • IVT
XYf cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
(c,0)
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx28
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-15 • 01Jul13% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m%% The Limitsxmin = 0; xmax1 = 3; xmin2 = xmax1; xmax = 3; ymin = -10; ymax = 15;% The FUNCTIONx1 = 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 appendingx = [x1, x2]; y = [y1, y2];% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onset(gca,'XTick',[xmin:1:xmax]); set(gca,'YTick',[ymin:5:ymax])hold off
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx29
Bruce Mayer, PE Chabot College Mathematics
WhiteBoard Work
Problems From §1.6• P13 → Find Limit Using Algebra• P52 → Electrically Charged Sphere• P56 → Create Continuity
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx30
Bruce Mayer, PE Chabot College Mathematics
All Done for Today
KnowYourLimits
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx31
Bruce Mayer, PE Chabot College Mathematics
Bruce Mayer, PELicensed Electrical & Mechanical Engineer
Chabot Mathematics
Appendix
–
srsrsr 22
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx32
Bruce Mayer, PE Chabot College Mathematics
Make Continuous - P
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx33
Bruce Mayer, PE Chabot College Mathematics
Make C
on
tinu
ou
s - Q
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx34
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx35
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx36
Bruce Mayer, PE Chabot College Mathematics
Charge Hollow Sphere E-fld
0 1 2 3-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
x/R
y =
E(x
) (V
olt/
me
ter)
MTH15 • Bruce Mayer, PE • P1.6-52 Charged Sphere
XYf cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx37
Bruce Mayer, PE Chabot College Mathematics
MA
TL
AB
Co
de
% Bruce Mayer, PE% MTH-15 • 01Jul13% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m%clear; clc;% InDep Var = x/R% The Limitsxmin = 0; xmax1 = 1; xmin2 = xmax1; xmax = 3; ymin = -.1; ymax = 1.1;% The FUNCTIONx1 = 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 appendingx = [x1, x2]; y = [y1, y2];% % The ZERO Lineszxh = [xmin xmax]; zyh = [0 0]; zxv = [0 0]; zyv = [ymin ymax];%% the 6x6 Plotaxes; set(gca,'FontSize',12);whitebg([0.8 1 1]); % Chg Plot BackGround to Blue-Greenplot(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 onplot(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
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx38
Bruce Mayer, PE Chabot College Mathematics
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx39
Bruce Mayer, PE Chabot College Mathematics
P1.6-52(B)
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx40
Bruce Mayer, PE Chabot College Mathematics
P1.6-56 Continuous Plot
0 1 2 3 4 5 6 7 8-50
-40
-30
-20
-10
0
10
x
f(x)
P
iece
Wis
tMTH15 • Bruce Mayer, PE • P1.6-56 PcWise Continuity
XYf cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m
[email protected] • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx41
Bruce Mayer, PE Chabot College Mathematics