Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

BMayer@ChabotCollege.edu • MTH15_Lec-06_sec_1-6_Limits-n_Continuity_.pptx1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

§1.6 Limits&

Continuity

Review §

Any QUESTIONS About• §1.5 → Limits

Any QUESTIONS About HomeWork• §1.5 → HW-05

§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

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!

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)

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

Continuous vs. DisContinuous

CONTINUOUS Function Plot

DIScontinuous Function Plot

Smooth vs. Kinked/Cornered

SMOOTH-Curved Function Plot

SHARP-Cornered Function Plot

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

X: 1.5Y: 1.034

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

ONEsided Limits – From RIGHT If f(x) Approaches L

as x→c from the Left; i.e., x<c, write:

xfx clim

-1 0 1 2 3 4-1

X: 1.5Y: 1.034

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

034.12

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)(

-3 -2 -1 0 1 2 3-4

MTH15 • Bruce Mayer, PE • 2-Sided Limit

XYf cnGraphBlueGreenBkGndSolidMarkerTemplate1306.m

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

41)1(3)13(lim)(lim11

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

xfxfxx

lim40lim

xfxfxx

limlim

% 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

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

limlim

Example Continuity Consider Function:

Determine if the Function is Continuous at• x = 4• x = 5

Use BiLateral Approach Limit Test

0 1 2 3 4 5 6 7 8 9 10-1000

MTH15 • Bruce Mayer, PE • Continuity Analysis

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

34327lim&

34327lim

234.979

59999.3

3439999.327

235.021

50001.4

3430001.427

343427

34327lim

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

MTH15 • Bruce Mayer, PE • Continuity Analysis

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

34327lim

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

eMTH15 • Bruce Mayer, PE • PcWise Continuous

Example Make Continuous

Consider the Fcn: This Fcn is

NONcontinuous asshown in the Plot

Make this Plot Continuous for Constants P & Q:

1if1153 2

1153 2

P-3 -2 -1 0 1 2 3

MTH15 • Bruce Mayer, PE • DIScontinuous

Example Continuous at 8 The FineTuned Fcn The

1if11524 2

-2 -1 0 1 2 3-15

Example Continuous at −13 The FineTuned Fcn The

1if7813

1if1152

-2 -1 0 1 2 3-20

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

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

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

f(c) = L

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

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

x2 - 2

MTH15 • Bruce Mayer, PE • IVT

% 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

WhiteBoard Work

Problems From §1.6• P13 → Find Limit Using Algebra• P52 → Electrically Charged Sphere• P56 → Create Continuity

All Done for Today

KnowYourLimits

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

BMayer@ChabotCollege.edu

Chabot Mathematics

Appendix

srsrsr 22

Make Continuous - P

Make C

Charge Hollow Sphere E-fld

0 1 2 3-0.1

MTH15 • Bruce Mayer, PE • P1.6-52 Charged Sphere

% 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

P1.6-52(B)

P1.6-56 Continuous Plot

0 1 2 3 4 5 6 7 8-50

tMTH15 • Bruce Mayer, PE • P1.6-56 PcWise Continuity

Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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