Limits and Continuity/Partial Derivatives

[Pages:47]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 )(x0,y0)

if for every > 0 there is a > 0 such that: for all (x, y ) in the domain of f if

0 < (x - x0)2 + (y - y0)2 <

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 )(x0,y0)

if for every > 0 there is a > 0 such that: for all (x, y ) in the domain of f if

0 < (x - x0)2 + (y - y0)2 <

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 +, -, ?, ?. 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 +, -, ?, ?. See for

example page 775 for a list.

For example:

if

lim f (x, y ) = L and lim g (x, y ) = K

(x,y )(x0,y0)

(x,y )(x0,y0)

Christopher Croke

Calculus 115

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

satisfies the same rules with respect to +, -, ?, ?. See for

example page 775 for a list.

For example:

if

lim f (x, y ) = L and lim g (x, y ) = K

(x,y )(x0,y0)

(x,y )(x0,y0)

then

lim f (x, y ) + g (x, y ) = L + K

(x,y )(x0,y0)

x2 + 2y + 3

lim

=?

(x,y )(1,0) x + y

Christopher Croke

Calculus 115

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

satisfies the same rules with respect to +, -, ?, ?. See for

example page 775 for a list.

For example:

if

lim f (x, y ) = L and lim g (x, y ) = K

(x,y )(x0,y0)

(x,y )(x0,y0)

then

lim f (x, y ) + g (x, y ) = L + K

(x,y )(x0,y0)

x2 + 2y + 3

lim

=?

(x,y )(1,0) x + y

x2 - y2

lim

=?

(x,y )(0,0) x - y

Christopher Croke

Calculus 115

Continuity(The definition is really the same.)

f is continuous at (x0, y0) if lim f (x, y ) = f (x0, y0).

(x,y )(x0,y0)

This means three things:

Christopher Croke

Calculus 115

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