THE EPSILON-DELTADEFINITIONOF A LIMIT

In the beginning …

In the beginning …

Leibniz and Newton created/discovered Calculus

Calculus soon became the mathematicsof change.

Calculus studied rates of change (derivatives)and accumulated change (integrals).

However, the foundations rested originallyon arguments that were very shaky.

Newton and Leibniz talked in terms ofinfinitesimals.

An infinitesimal is supposed to be a number that is greater than zero, butless than any positive number.

DUH?

Nonetheless, this illogic worked wellin practice.

To find the rate of change of the functionf(x)=x2, move an infinitesimal distance dxaway from x, and find the slope of the linethrough (x, f(x)) and (x+dx, f(x+dx)).

2 2 2 2 2

2

( )

2

( ) ( ) 2( )

2 (2 )

f x dx f x x dx x x xdx d

x dx

x xslopex dx x dx dx

xdx dx dx x dxdx dx

+ − + − + + −= = =

−

+= = = +

+

+

2 2 2 2 2

2

( )

2

( ) ( ) 2( )

2 (2 )

f x dx f x x dx x x xdx d

x dx

x xslopex dx x dx dx

xdx dx dx x dxdx dx

+ − + − + + −= = =

−

+= = = +

+

+

Since dx is infinitesimally small, discard itand you get the correct result that the rateof change is 2x.

In the 1700s, the theologian and philosopher Bishop Berkeley publisheda scathing critique of the illogical methods of calculus.

THEANALYST

OR, ADISCOURSE

Addressed to anInfidel Mathematician,

WHEREINIt is examined whether the Object, Principles, and

inferences of the modern analysis are more distinctlyconceived, or more evidently deduced, than

religious mysteries and points of faith.

THE ANALYST pointed to a need to develop a foundation for calculus that didnot depend on infinitesimals.

To meet this need, the concept of a limitwas formulated.

Instead of saying that we take a point that is infinitesimally close to x, we simply askwhat happens as we let values getarbitrarily close to x.

In particular, we often want to know whether f(x) gets closer to anything as our input values get closer to x. If so,then we call the value that f(x) gets closeto the limit.

However, the notion of “close,” whilemaybe intuitively clear, is not reallywell defined in the previous statement.

For a good definition of limit and what wemean by close, we need the epsilon-deltadefinition created by Karl Weierstrass in the 1800s.

Karl Weierstrass is known as the father ofmodern analysis. He started out as a high school teacher, but was later elevatedto professor at the Technical Universityof Berlin as a result of the many brilliant papers he published.

He defined “closeness” in terms of beingwithin a distance epsilon from somenumber.

In particular, we might paraphrase themodern definition of a limit as follows:

In particular, we might paraphrase themodern definition of a limit as follows:

To say that the limit of f(x) as x approachesa is equal to L means that we can make thevalue of f(x) within a distance of epsilonunits from L simply by making x within anappropriate distance of delta units from x.

We write this more formally as follows:

lim ( ) 0, 0

0 , ( ) .x a

f x L if and only if for every thereexists

suchthat if x a then f x L

ε δ

δ ε→

= > >

< − < − <

When I was young, we wrote this definition with even more mathematicalformalism:

lim ( ) 0, 0 0 ( ) .x a

f x L x a f x Lε δ δ ε→

= ⇔∀ > ∃ > ∋ < − < ⇒ − <

In this highly symbolic form, the upside downcapital “A” means “for every,” and the backwards “E” means “there exists.”

lim ( ) 0, 0 0 ( ) .x a

f x L x a f x Lε δ δ ε→

= ⇔∀ > ∃ > ∋ < − < ⇒ − <

So what does this actually mean?

So what does this actually mean?

It means that if we specify any positive numberepsilon, then we can always find a sufficientlysmall number delta such that if the distancebetween x and a is less than delta but greater than zero, then the distance between f(x) and Lwill be less than epsilon. If this works no matterhow small we make epsilon, then L is the limit ofour function as x approaches a.

Why do we require the distance betweenx and a to be greater than zero?

Why do we require the distance betweenx and a to be greater than zero?

Recall that the whole purpose of limits is to letus more rigorously define things such as thederivative.

In modern notation we write:

( ) ( )( ) limx a

f x f af xx a→

−′ =−

In modern notation we write:

( ) ( )( ) limx a

f x f af xx a→

−′ =−

If we let x equal a in this expression, then weare dividing by zero, and the universe as weknow it ceases to exist.

Thus, in the concept of the limit as x goes to a,we always want to consider what happens as x gets close to a while remaining different froma.

This practice also works well in those instanceswhere a function may not even be defined at a,or in cases where the function value at a may be different from the limit value.

1

1lim1x

xx→

−−

2 0( )

2 0if x

f xif x

≠⎧= ⎨− =⎩

KEY THEOREMS ABOUT LIMITS

lim ( ) lim ( ) . ,x a x a

Suppose f x L and g x M Then→ →

= =

( )( )( )( )( )

1. lim ( ) ( )

2. lim ( ) ( )

3. lim ( ) ( )

4. lim ( ) / ( ) / , 0

5. lim ( ) ,

x a

x a

x a

x a

x a

f x g x L M

f x g x L M

f x g x L M

f x g x L M provided M

c f x c M wherecis a fixed value

→

→

→

→

→

+ = +

− = −

⋅ = ⋅

= ≠

⋅ = ⋅

CONTINUITY

A function f(x) is continuous at a if:

lim ( ) ( )x a

f x f a→

=

CONTINUITY

A funcion f(x) is continuous at a if:

lim ( ) ( )x a

f x f a→

=

This is sometimes written as:

1. ( ) .2. lim ( ) .

3. lim ( ) ( ).x a

x a

f a is definedf x exists

f x f a→

→=

KEY THEOREMS ABOUT CONTINUITY

If f and g are continuous at a and if c is a constant, then the following functions are also continuous at a.

1.2.3.4. / , ( ) 05.

f gf gf gf g if g ac f

+−⋅

≠⋅

LIMITS FOR FUNCTIONS OF TWO VARIABLES

Definition: The disk D with center (a,b) and radius r is defined as:

( ) ( ){ }2 2( , ) :D x y x a y b r= − + − ≤

LIMITS FOR FUNCTIONS OF TWO VARIABLES

Let f be a function of two variables defined ona disk D with center (a,b). Then,

( ) ( )( , ) ( , )

2 2

lim ( , ) 0 0

0 , ( , ) .

x y a bf x y L if and only if for every thereexists a

suchthat if x a y b then f x y L

ε δ

δ ε

→= > >

< − + − < − <

The function f is continuous at (a,b) if and only if

( , ) ( , )lim ( , ) ( , )

x y a bf x y f a b

→=

EXAMPLES:

2 2

2 2( , ) ( , )lim ?

x y o o

x yx y→

−=

+

Approach (0,0) first along the x-axis and thenthe y-axis.

EXAMPLES:

2 2( , ) ( , )lim ?

x y o o

x yx y→

⋅=

+

Approach (0,0) first along the x-axis, thenalong the y-axis, and finally along the line y=x.

EXAMPLES:

2

4 2( , ) ( , )lim ?

x y o o

x yx y→

⋅=

+

Approach (0,0) first along the x-axis, thenalong the y-axis, then along any line y=mx,and finally along the parabola y=x2.

Prove:2

2 2( , ) ( , )

3lim 0x y o o

x yx y→

=+

Proof: 0 . ,3

Let and let Thenεε δ> =

( ) ( )2 2 2 20 0 03

x y x y εδ< − + − = + < = ⇒

22 2

2 2 2 2 2 2

2 2 2

33 30 3

3 3 3 3 .3

x yx y x y yx y x y x y

y x y εδ ε

− = = ≤+ + +

⎛ ⎞= ≤ + < = =⎜ ⎟⎝ ⎠

Prove:2

2 2( , ) ( , )

3lim 0x y o o

x yx y→

=+

Proof: 0 . ,3

Let and let Thenεε δ> =

( ) ( )2 2 2 20 0 03

x y x y εδ< − + − = + < = ⇒

22 2

2 2 2 2 2 2

2 2 2

33 30 3

3 3 3 3 .3

x yx y x y yx y x y x y

y x y εδ ε

− = = ≤+ + +

⎛ ⎞= ≤ + < = =⎜ ⎟⎝ ⎠

Therefore,2

2 2( , ) ( , )

3lim 0x y o o

x yx y→

=+

Is f(x,y) continuous at (2,3)?

2 2( , )f x y x y= +

Is f(x,y) continuous at (0,0)?

2 2

2 2 ( , ) (0,0)( , )

0 ( , ) (0,0)

x y if x yf x y x y

if x y

⎧ −≠⎪= +⎨

⎪ =⎩

Is f(x,y) continuous at (0,0)?

2 2 ( , ) (0,0)( , )1 ( , ) (0,0)x y if x yf x y

if x y⎧ + ≠⎪= ⎨

=⎪⎩