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
................
................
In order to avoid copyright disputes, this page is only a partial summary.
To fulfill the demand for quickly locating and searching documents.
It is intelligent file search solution for home and business.
Related download
- math 213 partial derivatives university of kentucky
- chapter 5 partial derivatives universiti teknologi malaysia
- lecture 9 partial derivatives harvard university
- unit 9 partial derivatives harvard university
- rules for finding derivatives whitman college
- section 14 3 partial derivatives with two variables
- partial derivatives university of pennsylvania
- partial derivatives
- limits and continuity partial derivatives
- partial derivatives examples and a quick review of implicit differentiation
Related searches
- derivatives of exponents and logs
- derivatives of logarithms and exponents
- second order partial derivatives calc
- four second partial derivatives calculator
- second partial derivatives calculator
- find the partial derivatives calculator
- partial derivatives khan academy
- partial derivatives explained
- limits and continuity calculator
- partial derivatives multivariable calculator
- second order partial derivatives calculator
- partial derivatives multivariable