SLecture2

MA 105 Calculus: Lecture 2Limits and Continuity

S. Sivaji Ganesh

Mathematics DepartmentIIT Bombay

July 27, 2009

S. Sivaji Ganesh (IIT Bombay) MA 105 Calculus: Lecture 2 July 27, 2009 1 / 27

Limits

DefinitionLet f be a function defined on both sides of a, except possibly at aitself.


Limits

DefinitionLet f be a function defined on both sides of a, except possibly at aitself. Then

limx→a

f (x) = L

and say ”the limit of f (x) as x approaches a, equals L“


Limits


limx→a

f (x) = L

and say ”the limit of f (x) as x approaches a, equals L“ if we can makethe values of f (x) arbitrarily close to L (as close to L as we like)


Limits


limx→a

f (x) = L

and say ”the limit of f (x) as x approaches a, equals L“ if we can makethe values of f (x) arbitrarily close to L (as close to L as we like) bytaking x to be sufficiently close to a (on either side of a)


Limits


limx→a

f (x) = L

and say ”the limit of f (x) as x approaches a, equals L“ if we can makethe values of f (x) arbitrarily close to L (as close to L as we like) bytaking x to be sufficiently close to a (on either side of a) but not equalto a.


Limits


limx→a

f (x) = L


Note The value of the function at a does not play any role in the abovedefinition.


Limits


limx→a

f (x) = L


Note The value of the function at a does not play any role in the abovedefinition. In fact, f need not be defined at a.See Appendix 1 for some pictures.


Limits

Compute1 lim

x→1

x−1x2−1


Limits

Compute1 lim

x→1

x−1x2−1 What is the domain of this function?


Limits

Compute1 lim

x→1

x−1x2−1 What is the domain of this function? What if my function

assigns 1 to 2009 ?


Limits

Compute1 lim

x→1


assigns 1 to 2009 ? Draw Graphs.


Limits

Compute1 lim

x→1


assigns 1 to 2009 ? Draw Graphs.2 lim

x→0

|x|x


Limits


Limits

Graph of Heaviside function H(x) =

{

0 if x < 0

1 if x ≥ 0limx→0 H(x) does

not exist. Why?


Limits

Graph of Heaviside function H(x) =

{

0 if x < 0

1 if x ≥ 0limx→0 H(x) does

not exist. Why?There is no L as in the definition of limit.


Limits

DefinitionLet f be a function defined on the left side (or both sides) of a, exceptpossibly at a itself.


Limits

DefinitionLet f be a function defined on the left side (or both sides) of a, exceptpossibly at a itself. Then

limx→a−

f (x) = L

and say ”the left-hand limit of f (x) as x approaches a, equals L“


Limits


limx→a−

f (x) = L

and say ”the left-hand limit of f (x) as x approaches a, equals L“ if wecan make the values of f (x) arbitrarily close to L (as close to L as welike)


Limits


limx→a−

f (x) = L

and say ”the left-hand limit of f (x) as x approaches a, equals L“ if wecan make the values of f (x) arbitrarily close to L (as close to L as welike) by taking x to be sufficiently close to a and x less than a.


Limits


limx→a−

f (x) = L


Note Once again, value of the function at a does not play any role inthe above definition.


Limits


limx→a−

f (x) = L


Note Once again, value of the function at a does not play any role inthe above definition. In fact, f need not be defined at a.


Limits


limx→a−

f (x) = L


Note Once again, value of the function at a does not play any role inthe above definition. In fact, f need not be defined at a. Similarlydefine ”the right-hand limit of f (x) as x approaches a, equals L“


Limits


limx→a−

f (x) = L


Note Once again, value of the function at a does not play any role inthe above definition. In fact, f need not be defined at a. Similarlydefine ”the right-hand limit of f (x) as x approaches a, equals L“ whichis denoted by limx→a+ f (x).

limx→0−

H(x) = 0 & limx→0+

H(x) = 1

.S. Sivaji Ganesh (IIT Bombay) MA 105 Calculus: Lecture 2 July 27, 2009 5 / 27

Limits


Limits

f (x) = sin(x)


Limits


Limits

f (x) =

−x if x ≤ 0

x if 0 < x < 4

−x + 8 if 4 < x < 8


Limits


Limits

f (x) =

{

−x if x < 0

ex if x ≥ 0


Limits

f (x) =√

x


Limits

Graphs of exp(x), log(x)


Limits



Limits


limx→a

f (x) = ∞

means that the values of f (x) can be made arbitrarily large (as large aswe like)


Limits


limx→a

f (x) = ∞

means that the values of f (x) can be made arbitrarily large (as large aswe like) by taking x to be sufficiently close to a (on either side of a)


Limits


limx→a

f (x) = ∞

means that the values of f (x) can be made arbitrarily large (as large aswe like) by taking x to be sufficiently close to a (on either side of a) butnot equal to a.


Limits


limx→a

f (x) = ∞


Note Once again, value of the function at a does not play any role inthe above definition.


Limits


limx→a

f (x) = ∞


Note Once again, value of the function at a does not play any role inthe above definition. In fact, f need not be defined at a.


Limits


limx→a

f (x) = ∞


Note Once again, value of the function at a does not play any role inthe above definition. In fact, f need not be defined at a.Similarly define limx→a f (x) = −∞.


Limits


limx→a

f (x) = ∞


Note Once again, value of the function at a does not play any role inthe above definition. In fact, f need not be defined at a.Similarly define limx→a f (x) = −∞.Go to the previous slide.


Limits



Limits

DefinitionLet f be a function defined on both sides of a, except possibly at aitself. Then The line x = a is called a vertical asymptote of the curvey = f (x) if at least one of the following statements is truelimx→a f (x) = ∞ limx→a− f (x) = ∞ limx→a+ f (x) = ∞limx→a f (x) = −∞ limx→a− f (x) = −∞ limx→a+ f (x) = −∞ .


Limits


Note Give some examples by drawing their graphs.


Limits


Note Give some examples by drawing their graphs.Go back and check if we already some examples.


Limits

f (x) = tan(x)


Limits

f (x) = tan(x)

Question List all the vertical asymptotes.


Limits

f (x) =1x2 , for x 6= 0


Limits

f (x) =1x

, for x 6= 0


Limits



Limits

DefinitionLet f be a function defined on both sides of a, except possibly at aitself. Then The line y = b is called a horizontal asymptote of the curvey = f (x) if


Limits

DefinitionLet f be a function defined on both sides of a, except possibly at aitself. Then The line y = b is called a horizontal asymptote of the curvey = f (x) if eitherlimx→∞ f (x) = b


Limits

DefinitionLet f be a function defined on both sides of a, except possibly at aitself. Then The line y = b is called a horizontal asymptote of the curvey = f (x) if eitherlimx→∞ f (x) = b or


Limits

DefinitionLet f be a function defined on both sides of a, except possibly at aitself. Then The line y = b is called a horizontal asymptote of the curvey = f (x) if eitherlimx→∞ f (x) = b or limx→−∞ f (x) = b holds.


Limits


Note Give some examples by drawing their graphs.


Limits


Note Give some examples by drawing their graphs.Go back and check if we already some examples.


Limits

f (x) = sin(1x

), for x > 0


Limits

f (x) = sin(1x

), for x > 0

Question Can you explain why limx→0+ f (x) does not exist?


Limit laws

We have seen some limit laws in the context of sequences. Similarlimit laws also hold for limits of functions.


Limit laws

We have seen some limit laws in the context of sequences. Similarlimit laws also hold for limits of functions.Let f , g be two functionsdefined on both sides of a, except possibly at a itself.


Limit laws

We have seen some limit laws in the context of sequences. Similarlimit laws also hold for limits of functions.Let f , g be two functionsdefined on both sides of a, except possibly at a itself.Assume thatlimx→a

f (x) and limx→a

g(x) exist. Then


Limit laws


f (x) and limx→a

g(x) exist. Then

1 limx→a

(f (x) + g(x)) = limx→a

f (x) + limx→a

g(x)


Limit laws


f (x) and limx→a

g(x) exist. Then

1 limx→a

(f (x) + g(x)) = limx→a

f (x) + limx→a

g(x)

2 limx→a

cf (x) = c limx→a

f (x) for any number c.


Limit laws


f (x) and limx→a

g(x) exist. Then

1 limx→a

(f (x) + g(x)) = limx→a

f (x) + limx→a

g(x)

2 limx→a

cf (x) = c limx→a


3 limx→a

f (x)g(x) = limx→a

f (x) limx→a

g(x)


Limit laws


f (x) and limx→a

g(x) exist. Then

1 limx→a

(f (x) + g(x)) = limx→a

f (x) + limx→a

g(x)

2 limx→a

cf (x) = c limx→a


3 limx→a


f (x) limx→a

g(x)

4 limx→a

1g(x) = 1

limx→a

g(x)


Limit laws


f (x) and limx→a

g(x) exist. Then

1 limx→a

(f (x) + g(x)) = limx→a

f (x) + limx→a

g(x)

2 limx→a

cf (x) = c limx→a


3 limx→a


f (x) limx→a

g(x)

4 limx→a

1g(x) = 1

limx→a

g(x) provided · · ·


Limit laws


f (x) and limx→a

g(x) exist. Then

1 limx→a

(f (x) + g(x)) = limx→a

f (x) + limx→a

g(x)

2 limx→a

cf (x) = c limx→a


3 limx→a


f (x) limx→a

g(x)

4 limx→a

1g(x) = 1

limx→a


Polynomials,


Limit laws


f (x) and limx→a

g(x) exist. Then

1 limx→a

(f (x) + g(x)) = limx→a

f (x) + limx→a

g(x)

2 limx→a

cf (x) = c limx→a


3 limx→a


f (x) limx→a

g(x)

4 limx→a

1g(x) = 1

limx→a


Polynomials,Rational functions,


Limit laws


f (x) and limx→a

g(x) exist. Then

1 limx→a

(f (x) + g(x)) = limx→a

f (x) + limx→a

g(x)

2 limx→a

cf (x) = c limx→a


3 limx→a


f (x) limx→a

g(x)

4 limx→a

1g(x) = 1

limx→a


Polynomials,Rational functions, all trigonometric functions whereverthey are defined, have property called direct substitution property:

limx→a

f (x) = f (a)


Limits

f (x) = x sin(1x

), for x 6= 0


Limits

f (x) = x sin(1x

), for x 6= 0

What is limx→0

x sin(1x )?


Limits

f (x) = x sin(1x

), for x 6= 0

What is limx→0

x sin(1x )? Why?


Limits

f (x) = x sin(1x

), for x 6= 0

What is limx→0

x sin(1x )? Why? How to prove?


Squeeze (Sandwich) Theorem for Limits

TheoremIf f (x) ≤ g(x) when x is near a (except possibly at a) and the limits of fand g both exist as x approaches a, then

limx→a

f (x) ≤ limx→a

g(x).




limx→a

f (x) ≤ limx→a

g(x).

The Squeeze Theorem

If f (x) ≤ g(x) ≤ h(x) when x is near a (except possibly at a) and




limx→a

f (x) ≤ limx→a

g(x).

The Squeeze Theorem


limx→a

f (x) = limx→a

h(x) = L,




limx→a

f (x) ≤ limx→a

g(x).

The Squeeze Theorem


limx→a

f (x) = limx→a

h(x) = L,

thenlimx→a

g(x) = L.




limx→a

f (x) ≤ limx→a

g(x).

The Squeeze Theorem


limx→a

f (x) = limx→a

h(x) = L,

thenlimx→a

g(x) = L.

Note that −x ≤ x sin(1/x) ≤ x




limx→a

f (x) ≤ limx→a

g(x).

The Squeeze Theorem


limx→a

f (x) = limx→a

h(x) = L,

thenlimx→a

g(x) = L.

Note that −x ≤ x sin(1/x) ≤ x and apply Squeeze theorem.


Limits

behaviour of sin(x) at infinity


Limits


Question Can you explain why limx→∞ sin(x) does not exist?


Limits


Question Can you explain why limx→∞ sin(x) does not exist? Did wesee this already?


Limits


Question Can you explain why limx→∞ sin(x) does not exist? Did wesee this already? Recall sin(n) exercise; solve it graphically.


ǫ-δ definition of Limit

Let f be a function defined on some



Let f be a function defined on some open interval



Let f be a function defined on some open interval that contains a,except possibly at a itself.



Let f be a function defined on some open interval that contains a,except possibly at a itself. Then we say that the limit of f (x) as xapproaches a is L and we write

limx→a

f (x) = L.




limx→a

f (x) = L.

if




limx→a

f (x) = L.

if for every ǫ > 0




limx→a

f (x) = L.

if for every ǫ > 0 there is a number δ > 0 such that

|f (x) − L| < ǫ whenever 0 < |x − a| < δ.




limx→a

f (x) = L.



Define the precise definitions of left-hand and right-hand limits.Play the ǫ-δ game in many examples,




limx→a

f (x) = L.



Define the precise definitions of left-hand and right-hand limits.Play the ǫ-δ game in many examples, including the next one.Also, see Appendix 2.


ǫ-δ definition of infinite imits

Let f be a function defined on some



Let f be a function defined on some open interval



Let f be a function defined on some open interval that contains a,except possibly at a itself.



Let f be a function defined on some open interval that contains a,except possibly at a itself. Then we write

limx→a

f (x) = ∞




limx→a

f (x) = ∞

if




limx→a

f (x) = ∞

if for every M > 0




limx→a

f (x) = ∞

if for every M > 0 there is a number δ > 0 such that

f (x) > M whenever 0 < |x − a| < δ.




limx→a

f (x) = ∞



Define limx→a f (x) = −∞




limx→a

f (x) = ∞



Define limx→a f (x) = −∞Prove that limx→0

1x2 = ∞.




limx→a

f (x) = ∞



Define limx→a f (x) = −∞Prove that limx→0

1x2 = ∞. What do we need to do?


Continuity

1 A function f is continuous from the right at a if


Continuity


limx→a+

f (x) = f (a).


Continuity


limx→a+

f (x) = f (a).

2 A function f is continuous from the left at a if


Continuity


limx→a+

f (x) = f (a).


limx→a−

f (x) = f (a).


Continuity


limx→a+

f (x) = f (a).


limx→a−

f (x) = f (a).

3 A function f is continuous at a if


Continuity


limx→a+

f (x) = f (a).


limx→a−

f (x) = f (a).


limx→a

f (x) = f (a)


Continuity


limx→a+

f (x) = f (a).


limx→a−

f (x) = f (a).


limx→a

f (x) = f (a)

4 A function f is continuous on an interval if


Continuity


limx→a+

f (x) = f (a).


limx→a−

f (x) = f (a).


limx→a

f (x) = f (a)

4 A function f is continuous on an interval if f is continuous at everynumber in the interval.


Continuity


limx→a+

f (x) = f (a).


limx→a−

f (x) = f (a).


limx→a

f (x) = f (a)

4 A function f is continuous on an interval if f is continuous at everynumber in the interval.In case f is defined only on one side of anend point of the interval,


Continuity


limx→a+

f (x) = f (a).


limx→a−

f (x) = f (a).


limx→a

f (x) = f (a)

4 A function f is continuous on an interval if f is continuous at everynumber in the interval.In case f is defined only on one side of anend point of the interval, continuous at the end point means


Continuity


limx→a+

f (x) = f (a).


limx→a−

f (x) = f (a).


limx→a

f (x) = f (a)

4 A function f is continuous on an interval if f is continuous at everynumber in the interval.In case f is defined only on one side of anend point of the interval, continuous at the end point meanscontinuous from the right or continuous from the left.


Continuity

Note that the definition for continuity of a function f at a is equivalent tothe following:


Continuity


1 The function f must be defined at a.


Continuity


1 The function f must be defined at a. i.e., a is in the domain of f ,


Continuity


1 The function f must be defined at a. i.e., a is in the domain of f ,2 limx→a f (x) exists,


Continuity


1 The function f must be defined at a. i.e., a is in the domain of f ,2 limx→a f (x) exists, and3 limx→a f (x) = L.

TheoremIf f and g are continuous at a, then the functions f + g,


Continuity



TheoremIf f and g are continuous at a, then the functions f + g, f − g,


Continuity



TheoremIf f and g are continuous at a, then the functions f + g, f − g, cg (c is aconstant),


Continuity



TheoremIf f and g are continuous at a, then the functions f + g, f − g, cg (c is aconstant), fg,


Continuity



TheoremIf f and g are continuous at a, then the functions f + g, f − g, cg (c is aconstant), fg, f/g (provided ....),


Continuity



TheoremIf f and g are continuous at a, then the functions f + g, f − g, cg (c is aconstant), fg, f/g (provided ....), f◦g (composition of f and g,


Continuity



TheoremIf f and g are continuous at a, then the functions f + g, f − g, cg (c is aconstant), fg, f/g (provided ....), f◦g (composition of f and g, wheneverit makes sense) are all continuous.


Continuity




Thus, Polynomials,


Continuity




Thus, Polynomials, Rational functions,


Continuity




Thus, Polynomials, Rational functions, trigonometric functions


Continuity




Thus, Polynomials, Rational functions, trigonometric functions are allcontinuous


Continuity




Thus, Polynomials, Rational functions, trigonometric functions are allcontinuous on their respective domains.


Properties of continuous functions



Take a continuous function on an interval. Take any two values in therange



Take a continuous function on an interval. Take any two values in therange and take any third value lying between them.



Take a continuous function on an interval. Take any two values in therange and take any third value lying between them. What can we sayabout it?



Take a continuous function on an interval. Take any two values in therange and take any third value lying between them. What can we sayabout it? Does it also belong to the range?




Intermediate value theoremSuppose that f is continuous on the closed interval [a, b] and




Intermediate value theoremSuppose that f is continuous on the closed interval [a, b] and let N beany number between f (a) and f (b), where f (a) 6= f (b). Then thereexists a c ∈ (a, b) such that f (c) = N.





Understand what is meant by between.





Understand what is meant by between.Verify the validity of the theorem by drawing graphs ofmanycontinuous functions.





Understand what is meant by between.Verify the validity of the theorem by drawing graphs ofmanycontinuous functions.Intersting question Is every continuous function given by aformula?





Understand what is meant by between.Verify the validity of the theorem by drawing graphs ofmanycontinuous functions.Intersting question Is every continuous function given by aformula?Roughly speaking,





Understand what is meant by between.Verify the validity of the theorem by drawing graphs ofmanycontinuous functions.Intersting question Is every continuous function given by aformula?Roughly speaking, a continuous function is





Understand what is meant by between.Verify the validity of the theorem by drawing graphs ofmanycontinuous functions.Intersting question Is every continuous function given by aformula?Roughly speaking, a continuous function is the one whose graphcan be drawn without





Understand what is meant by between.Verify the validity of the theorem by drawing graphs ofmanycontinuous functions.Intersting question Is every continuous function given by aformula?Roughly speaking, a continuous function is the one whose graphcan be drawn without lifting pen from the paper.





Understand what is meant by between.Verify the validity of the theorem by drawing graphs ofmanycontinuous functions.Intersting question Is every continuous function given by aformula?Roughly speaking, a continuous function is the one whose graphcan be drawn without lifting pen from the paper.Can we draw the graph of every continuous function?


about Intermediate Value Theorem



Proof is non-trivial



Proof is non-trivial and uses V.I.P. of Real numbers.




Formulate a converse of Intermediate value theorem




Formulate a converse of Intermediate value theorem and is ittrue?




Formulate a converse of Intermediate value theorem and is ittrue? Draw graphs and find out for yourself.





Applications of Intermediate Value Theorem

Helps us in deciding a given function is not continuous on aninterval.







For odd degree polynomials P(x) with real coefficients,







For odd degree polynomials P(x) with real coefficients, if we canshow that P takes positve as well as negative values,







For odd degree polynomials P(x) with real coefficients, if we canshow that P takes positve as well as negative values, we concludethat the polynomial P(x) has a real root.







For odd degree polynomials P(x) with real coefficients, if we canshow that P takes positve as well as negative values, we concludethat the polynomial P(x) has a real root. Prove







For odd degree polynomials P(x) with real coefficients, if we canshow that P takes positve as well as negative values, we concludethat the polynomial P(x) has a real root. Prove

See Appendix 3 also.


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