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

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Chabot Mathematics. §1.6 Limits &amp; Continuity. Bruce Mayer, PE Licensed Electrical &amp; Mechanical Engineer BMayer@ChabotCollege.edu. 1.5. 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

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Chabot Mathematics

§1.6 Limits& Continuity

Bruce Mayer, PE

1.5

Review §
• §1.5 → Limits
• §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
• DIScontinuousFunction 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
ONEsided Limits – From RIGHT
• If f(x) Approaches L as x→c from the Left; i.e., x<c, write:
• See Graph at Right
Example  PieceWiseFcn
• Find the OneSidedLimits for Function:
• Compute the one-sided limits of f(x) as xapproaches 1
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
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

% 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax];

%

% 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([xminxmaxyminymax]),...

grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrowPieceWise'),...

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

MATLAB Code
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
Example  Continuity
• Consider Function:
• See Graph at Right
• Determine if the Function is Continuous at
• x = 4
• x = 5
• Use BiLateral Approach Limit Test
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
Example  Continuity
• Now Check Continuity at x = 5
• Use Approach Tables
• From Approach Tables Note:
PieceWise Continuity
• A NONontinuousPieceWise-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
Example  Make Continuous
• Consider the Fcn:
• This Fcn isNONcontinuous asshown in the Plot
• Make this Plot Continuous for Constants P & Q:
Example  Continuous at 8
• The FineTunedFcn
• ThePlot
Example  Continuous at −13
• The FineTunedFcn
• ThePlot

% 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax];

%

% 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([xminxmaxyminymax]),...

grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrowPieceWise'),...

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

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 = -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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax];

%

% 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([xminxmaxyminymax]),...

grid, xlabel('\fontsize{14}x'), ylabel('\fontsize{14}f(x) \rightarrowPieceWise'),...

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

Q MATLAB Code
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

f(b)

f(c) = L

f(a)

a

c

b

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
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.

(c,0)

% 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax];

%

% 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([xminxmaxyminymax]),...

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

MATLAB Code
WhiteBoard Work
• Problems From §1.6
• P13 → Find Limit Using Algebra
• P52 → Electrically Charged Sphere
• P56 → Create Continuity
All Done for Today

KnowYourLimits

Chabot Mathematics

Appendix

Bruce Mayer, PE

% Bruce Mayer, PE

% MTH-15 • 01Jul13

% XY_fcn_Graph_BlueGreenBkGnd_Solid_Marker_Template1306.m

%

clear; clc;

% InDepVar = 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 = [xminxmax]; zyh = [0 0]; zxv = [0 0]; zyv = [yminymax];

%

% 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([xminxmaxyminymax]),...

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

MATLAB Code