Limits and Continuity/Partial Derivatives

Limits and Continuity/Partial Derivatives

Christopher Croke

University of Pennsylvania

Math 115

UPenn, Fall 2011

Christopher Croke

Calculus 115

Limits

For (x0 , y0 ) an interior or a boundary point of the domain of a

function f (x, y ).

Definition:

lim

f (x, y ) = L

(x,y )¡ú(x0 ,y0 )

if for every  > 0 there is a ¦Ä > 0 such that: for all (x, y ) in the

domain of f if

q

0 < (x ? x0 )2 + (y ? y0 )2 < ¦Ä

then

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

Christopher Croke

Calculus 115

Limits

For (x0 , y0 ) an interior or a boundary point of the domain of a

function f (x, y ).

Definition:

lim

f (x, y ) = L

(x,y )¡ú(x0 ,y0 )

if for every  > 0 there is a ¦Ä > 0 such that: for all (x, y ) in the

domain of f if

q

0 < (x ? x0 )2 + (y ? y0 )2 < ¦Ä

then

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

Christopher Croke

Calculus 115

This definition is really the same as in one dimension and so

satisfies the same rules with respect to +, ?, ¡Á, ¡Â. See for

example page 775 for a list.

Christopher Croke

Calculus 115

This definition is really the same as in one dimension and so

satisfies the same rules with respect to +, ?, ¡Á, ¡Â. See for

example page 775 for a list.

For example:

if

lim

f (x, y ) = L and

lim

g (x, y ) = K

(x,y )¡ú(x0 ,y0 )

(x,y )¡ú(x0 ,y0 )

Christopher Croke

Calculus 115

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