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We will be covering here:

limits of functions of one and two variables.

limits of functions of one and two variables.

1.
Limits

Functions of one

and Two

Variables

Functions of one

and Two

Variables

2.
Limits for Functions of One

Variable.

What do we mean when we say that

lim f ( x) L?

x a

Informally, we might say that as x gets “closer and

closer” to a, f(x) should get “closer and closer” to L.

This informal explanation served pretty well in

beginning calculus, but in order to extend the idea to

functions of several variables, we have to be a bit more

precise.

Variable.

What do we mean when we say that

lim f ( x) L?

x a

Informally, we might say that as x gets “closer and

closer” to a, f(x) should get “closer and closer” to L.

This informal explanation served pretty well in

beginning calculus, but in order to extend the idea to

functions of several variables, we have to be a bit more

precise.

3.
Defining the Limit

Remember: the pt. lim f ( x) L

(a,f(a)) is excluded! x a

Means that

•given any tolerance T for L

L+T • we can find a tolerance t for

a

L such that

L-T •if x is between a-t and a+t,

but x is not a,

• f(x) will be between L-T and

L+T.

(Graphically, this means that the

part of the graph that lies in the

yellow vertical strip---that is,

those values that come from

a-t a a+t (a-t,a+t)--- will also lie in the

orange horizontal strip.)

Remember: the pt. lim f ( x) L

(a,f(a)) is excluded! x a

Means that

•given any tolerance T for L

L+T • we can find a tolerance t for

a

L such that

L-T •if x is between a-t and a+t,

but x is not a,

• f(x) will be between L-T and

L+T.

(Graphically, this means that the

part of the graph that lies in the

yellow vertical strip---that is,

those values that come from

a-t a a+t (a-t,a+t)--- will also lie in the

orange horizontal strip.)

4.
This isn’t True for This

function!

No amount of making the

Tolerance around a smaller is

going to force the graph of

L+T that part of the function

within the bright orange

L strip!

a

lim f ( x) L

x a

function!

No amount of making the

Tolerance around a smaller is

going to force the graph of

L+T that part of the function

within the bright orange

L strip!

a

lim f ( x) L

x a

5.
Changing the value of L doesn’t help

either!

L

a

In fact, there is no L that will help us. lim f ( x) does not exist.

x a

either!

L

a

In fact, there is no L that will help us. lim f ( x) does not exist.

x a

6.
Functions of Two Variables

How does this extend to functions of two variables?

We can start with informal language as before:

lim f ( x, y ) L

( x , y ) ( a ,b )

means that as (x,y) gets “closer and closer” to (a,b) , f(x,y)

gets closer and closer to L.

How does this extend to functions of two variables?

We can start with informal language as before:

lim f ( x, y ) L

( x , y ) ( a ,b )

means that as (x,y) gets “closer and closer” to (a,b) , f(x,y)

gets closer and closer to L.

7.
“Closer and Closer”

The words “closer and closer” obviously have to do with

measuring distance.

In the real numbers, one number is “close” to another if it is

within a certain tolerance---say no bigger than a+.01 and no

smaller than a-.01.

In the plane, one point is “close” to another if it is within a

certain fixed distance---a radius!

r

(x,y)

(a,b)

The words “closer and closer” obviously have to do with

measuring distance.

In the real numbers, one number is “close” to another if it is

within a certain tolerance---say no bigger than a+.01 and no

smaller than a-.01.

In the plane, one point is “close” to another if it is within a

certain fixed distance---a radius!

r

(x,y)

(a,b)

8.
What about those strips?

The vertical strip

becomes a

r

(a,b) (x,y)

The vertical strip

becomes a

r

(a,b) (x,y)

9.
Horizontal Strip?

L lies on the z-axis. Remember that the

We are interested function values are

in function values L+T back in the real

that lie between L numbers, so

z=L-T and z=L+T L-T “closeness” is once

again measured in

terms of

“tolerance.”

The set of all z-values

that lie between L-T

and L+T, are

“trapped” between

the two horizontal

planes z=L-T and The horizontal strip

becomes a

“sandwich”!

L lies on the z-axis. Remember that the

We are interested function values are

in function values L+T back in the real

that lie between L numbers, so

z=L-T and z=L+T L-T “closeness” is once

again measured in

terms of

“tolerance.”

The set of all z-values

that lie between L-T

and L+T, are

“trapped” between

the two horizontal

planes z=L-T and The horizontal strip

becomes a

“sandwich”!

10.
Putting it All Together

The part of the graph that

lies above the green circle

must also lie between the

two horizontal planes.

lim f ( x, y ) L

( x , y ) ( a ,b )

if given any pair of horizontal

planes about L, we can find a

circle centered at (a,b) so that the

part of the graph of f within the

cylinder is also between the

The part of the graph that

lies above the green circle

must also lie between the

two horizontal planes.

lim f ( x, y ) L

( x , y ) ( a ,b )

if given any pair of horizontal

planes about L, we can find a

circle centered at (a,b) so that the

part of the graph of f within the

cylinder is also between the

11.
Defining the Limit

lim f ( x, y ) L

( x , y ) ( a ,b )

L+T

Means that

L •given any tolerance T for L

L-T • we can find a radius r about

(a,b)

such that

•if (x,y) lies within a distance r

from (a,b), with (x,y) different

from (a,b) ,

• f(x,y) will be between L-T

and L+T.

Once again, the pt. ((a,b), f(a,b)) can be anywhere (or nowhere) !

lim f ( x, y ) L

( x , y ) ( a ,b )

L+T

Means that

L •given any tolerance T for L

L-T • we can find a radius r about

(a,b)

such that

•if (x,y) lies within a distance r

from (a,b), with (x,y) different

from (a,b) ,

• f(x,y) will be between L-T

and L+T.

Once again, the pt. ((a,b), f(a,b)) can be anywhere (or nowhere) !