M38 Lec 020414

MATH 38Mathematical Analysis III

I. F. Evidente

IMSP (UPLB)

Outline

1 Limits of Functions of Two or More VariablesLimits Along PathsIntuitive Notion of LimitsBasic Limit TheoremsShowing the nonexistence of limitsLimits: Formal Definition

2 Continuity

Figures taken from: J. Stewart, The Calculus: Early Transcendentals,Brooks/Cole, 6th Edition, 2008.

DefinitionA function of two variables f (x, y) is a rule that assigns a unique realnumber to each point (x, y) in some subset D of R2.

RemarkThe graph of f (x, y) is the surface in R3 (Euclidean 3-space) satisfyingthe equation z = f (x, y).

RemarkSuppose is the graph of f . Geometrically:

The domain of f is Projxy ()

The range of f is the "projection" of onto the z-axis.

RemarkSuppose is the graph of f . Geometrically:

The domain of f is Projxy ()The range of f is the "projection" of onto the z-axis.

Example

Use the graph of f to determine its domain and range:1 f (x, y)= 4x2+ y22 f (x, y)=

9x2 y2

Examplef (x, y)= 4x2+ y2

Example

f (x, y)=9x2 y2

Outline


2 Continuity

Limits Along PathsLet f (x, y) be a function of two variables.

Let C be a smooth curve on thexy-plane with equation y = g (x) whose points are in the domain of f . Let(x0, y0) be a point on C . The limit of f (x, y) as (x, y) approaches thepoint (x0, y0) along the curve C is:

lim(x,y)(x0,y0)

f (x, y)= limxx0

f (x,g (x))

In the case that C has equation x = g (y), then

lim(x,y)(x0,y0)

f (x, y)= limxy0

f (g (y), y)

Limits Along PathsLet f (x, y) be a function of two variables. Let C be a smooth curve on thexy-plane with equation y = g (x) whose points are in the domain of f .

Let(x0, y0) be a point on C . The limit of f (x, y) as (x, y) approaches thepoint (x0, y0) along the curve C is:

lim(x,y)(x0,y0)

f (x, y)= limxx0

f (x,g (x))


lim(x,y)(x0,y0)

f (x, y)= limxy0

f (g (y), y)

Limits Along PathsLet f (x, y) be a function of two variables. Let C be a smooth curve on thexy-plane with equation y = g (x) whose points are in the domain of f . Let(x0, y0) be a point on C .

The limit of f (x, y) as (x, y) approaches thepoint (x0, y0) along the curve C is:

lim(x,y)(x0,y0)

f (x, y)= limxx0

f (x,g (x))


lim(x,y)(x0,y0)

f (x, y)= limxy0

f (g (y), y)

Limits Along PathsLet f (x, y) be a function of two variables. Let C be a smooth curve on thexy-plane with equation y = g (x) whose points are in the domain of f . Let(x0, y0) be a point on C . The limit of f (x, y) as (x, y) approaches thepoint (x0, y0) along the curve C is:

lim(x,y)(x0,y0)

f (x, y)= limxx0

f (x,g (x))


lim(x,y)(x0,y0)

f (x, y)= limxy0

f (g (y), y)

Examples

Example

1 Find lim(x,y)(0,0)

x2

x2+ y2 along the path y = 0, x = 0 and y = x.

2 Find lim(x,y)(0,0)

x9y

(x6+ y2)2 along the path y = x and y = x2.

RecallFor functions of one variable:

There two ways of approaching x0: from the left and from the rightConcept of one-sided limits: only two typesSince the limit is the unique value that the y-values approach as thex-values approach x0 from both sides, lim

xa f (x) exists if and only iflimxa+

f (x)= limxa f (x)


There two ways of approaching x0: from the left and from the right

Concept of one-sided limits: only two typesSince the limit is the unique value that the y-values approach as thex-values approach x0 from both sides, lim


f (x)= limxa f (x)


There two ways of approaching x0: from the left and from the rightConcept of one-sided limits: only two types

Since the limit is the unique value that the y-values approach as thex-values approach x0 from both sides, lim


f (x)= limxa f (x)


There two ways of approaching x0: from the left and from the rightConcept of one-sided limits: only two typesSince the limit is the unique value that the y-values approach as thex-values approach x0 from both sides, lim


f (x)= limxa f (x)

For functions of two variables:

Intuitive Notion of LimitsFor functions of two variables:

The point (x, y) may approach (x0, y0) via an infinite number of paths.Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!This makes the computation of limits of functions of two variablesmuch more complex.We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.


The point (x, y) may approach (x0, y0) via an infinite number of paths.

Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!This makes the computation of limits of functions of two variablesmuch more complex.We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.


The point (x, y) may approach (x0, y0) via an infinite number of paths.Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!

This makes the computation of limits of functions of two variablesmuch more complex.We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.


The point (x, y) may approach (x0, y0) via an infinite number of paths.Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!This makes the computation of limits of functions of two variablesmuch more complex.

We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.


The point (x, y) may approach (x0, y0) via an infinite number of paths.Intuitively, the limit of f (x, y) as (x, y) approaches (x0, y0) is theunique value that f (x, y) approaches as (x, y) approaches (x0, y0) viaall these possible paths!This makes the computation of limits of functions of two variablesmuch more complex.We restrict our computation of limits to very simple types offunctions: polynomial and rational functions.

Theorem1 lim

(x,y)(x0,y0)c = c

2 If f (x, y) is a polyomial function, then lim(x,y)(x0,y0)

f (x, y)= f (x0, y0)

Examples

1 lim(x,y)(1,2)

2x3y =

2 13 2= 22 lim

(x,y)(0,0)1000= 1000

Examples

1 lim(x,y)(1,2)

2x3y = 2 13 2=

22 lim

(x,y)(0,0)1000= 1000

Examples

1 lim(x,y)(1,2)

2x3y = 2 13 2= 2

2 lim(x,y)(0,0)

1000= 1000

Examples

1 lim(x,y)(1,2)

2x3y = 2 13 2= 22 lim

(x,y)(0,0)1000=

1000

Examples

1 lim(x,y)(1,2)

2x3y = 2 13 2= 22 lim

(x,y)(0,0)1000= 1000

Theorem (Properties of Limits)If lim

(x,y)(x0,y0)f (x, y)= L and lim

(x,y)(x0,y0)g (x, y)=M , then

1 lim(x,y)(x0,y0)

f (x, y) g (x, y)= LM2 If c R, then lim

(x,y)(x0,y0)c f (x, y)= c L.

3 lim(x,y)(x0,y0)

f (x, y) g (x, y)= L M .

4 If M 6= 0, then lim(x,y)(x0,y0)

f (x, y)

g (x, y)= LM

.

Remark

As a consequence of the last two theorems, if h(x, y)= f (x, y)g (x, y)

is a rational

function where f and g are polynomial functions,

lim(x,y)(x0,y0)

h(x, y)= f (x0, y0)g (x0, y0)

provided that g (x0, y0) 6= 0.

Examples

1 lim(x,y)(0,0)

x22y +2x y +1

2 lim(x,y)(0,0)

x2

x2+ y2

3 lim(x,y)(0,0)

x4 y4x2+ y2

RemarkLet f (x, y) be a function of two variables and let C1 and C2 be differentpaths in the domain of f containing the point (x0, y0).

If the limits off (x, y) as (x, y) approaches the point (x0, y0) along the curves C1 and C2are not equal, then the limit of f (x, y) as (x, y) approaches the point(x0, y0) does not exist.

RemarkLet f (x, y) be a function of two variables and let C1 and C2 be differentpaths in the domain of f containing the point (x0, y0). If the limits off (x, y) as (x, y) approaches the point (x0, y0) along the curves C1 and C2are not equal, then

the limit of f (x, y) as (x, y) approaches the point(x0, y0) does not exist.

RemarkLet f (x, y) be a function of two variables and let C1 and C2 be differentpaths in the domain of f containing the point (x0, y0). If the limits off (x, y) as (x, y) approaches the point (x0, y0) along the curves C1 and C2are not equal, then the limit of f (x, y) as (x, y) approaches the point(x0, y0) does not exist.

Standard Curvesx = 0, y = 0, y = x, y = x2, y = x3

ExampleShow that the given limit does not exist:

1 lim(x,y)(0,0)

x2

x2+ y2

2 lim(x,y)(0,0)

x9y

(x6+ y2)2

3 lim(x,y)(0,1)

x(y 1)3x5x2+3(y 1)2

DistanceLet P and Q be points in 2- or 3-space. We denote the distance d betweenP and Q by d = ||P Q||.

1 2-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2


1 2-space:

d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2


1 2-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2


1 2-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space:

d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2


1 2-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2

2 3-space: d = ||P Q|| =(x1x2)2+ (y1 y2)2+ (z1 z2)2

DefinitionIf P R2 or R3 and r > 0, then the open ball centered at P and of radiusr is

B(P,r )= {Q R2 | ||P Q|| < r }

DefinitionLet P R2 and let f be a function in 2 variables defined on some open ballB(P,r ) centered at P , except possibly at P . Then

lim(x,y)(x0,y0)

f (x, y)= f (x0, y0)

if the following condition is satisfied:For every > 0, there exists > 0 such that if

(xx0)2+ (y y0)2 < ,

then | f (x, y)L| < .

For every > 0, there exists > 0 such that if(xx0)2+ (y y0)2 < ,

then | f (x, y)L| <

Outline


2 Continuity

DefinitionThe function f (x, y) is continuous at P = (x0, y0) if and only if

lim(x,y)(x0,y0)

f (x, y)= f (x0, y0).

ExampleDetermine whether the given function is continuous at (0,0).

1 f (x, y)= x22y +2x y +1

2 f (x, y)= x4 y4

x2+ y2

3 f (x, y)= x2

x2+ y24 f (x, y)= 1

x2+ y2

5 f (x, y)=

x4 y4x2+ y2 , (x, y) 6= (0,0)(0,0), (x, y)= (0,0)

Types of Discontinuities1 Removable: limit exists, but is not equal to the function value2 Essential: limit does not exist

ExampleDetermine the type of discontinuity at (0,0):

1 f (x, y)= x4 y4

x2+ y2

(removable)

2 f (x, y)= x2

x2+ y2

(essential)

3 f (x, y)= 1x2+ y2

(essential)


1 f (x, y)= x4 y4

x2+ y2 (removable)

2 f (x, y)= x2

x2+ y2

(essential)

3 f (x, y)= 1x2+ y2

(essential)


1 f (x, y)= x4 y4

x2+ y2 (removable)

2 f (x, y)= x2

x2+ y2 (essential)

3 f (x, y)= 1x2+ y2

(essential)


1 f (x, y)= x4 y4

x2+ y2 (removable)

2 f (x, y)= x2

x2+ y2 (essential)

3 f (x, y)= 1x2+ y2 (essential)

f (x, y)= x22y +2x y +1 (continuous)

f (x, y)= x4 y4

x2+ y2 (removable)

f (x, y)= 1x2+ y2 (essential)

f (x, y)= x2

x2+ y2 (essential)

DefinitionA function of 2 variables that is continuous at every point (x, y) is said tobe continuous everywhere, or simply continuous.

Theorem (Continuity Theorems)1 A polynomial function is continuous everywhere.

2 A rational function is continuous at every point in its domain.3 A sum, difference, product, quotient (except where the denominator is

0) or composition of continuous functions are continuous.

Theorem (Continuity Theorems)1 A polynomial function is continuous everywhere.2 A rational function is continuous at every point in its domain.

3 A sum, difference, product, quotient (except where the denominator is0) or composition of continuous functions are continuous.

Theorem (Continuity Theorems)1 A polynomial function is continuous everywhere.2 A rational function is continuous at every point in its domain.3 A sum, difference, product, quotient (except where the denominator is

0) or composition of continuous functions are continuous.

TheoremIf lim

(x,y)(x0,y0)g (x, y)= b and f is a function of one variable that is

continuous at b, then lim(x,y)(x0,y0)

( f g )(x, y)= f(

lim(x,y)(x0,y0)

g (x, y)

)= f (b)

ExampleEvaluate the following limits:

1 lim(x,y)(pi,0)

cos(x+ y)2 lim

(x,y)(e,1)ln(xy2)

Limits of Functions of Two or More VariablesLimits Along PathsIntuitive Notion of LimitsBasic Limit TheoremsShowing the nonexistence of limitsLimits: Formal Definition

Continuity

M38 Lec 020414

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