Calculus (I)
WEN-CHING LIEN
Department of Mathematics National Cheng Kung University
2008
14.2 Limits and Continuity
Definition
An open disk with radius r centered at(x0,y0)∈R2 is the set
Br(x0,y0) ={(x,y)∈R2:p
(x−x0)2+ (y −y0)2<r}
A closed disk with radius r centered at(x0,y0)∈R2 is the set
B¯(x ,y ) ={(x,y)∈R2 :p
(x−x )2+ (y −y )2≤r}
14.2 Limits and Continuity
Definition
An open disk with radius r centered at(x0,y0)∈R2 is the set
Br(x0,y0) ={(x,y)∈R2:p
(x−x0)2+ (y −y0)2<r}
A closed disk with radius r centered at(x0,y0)∈R2 is the set
B¯(x ,y ) ={(x,y)∈R2 :p
(x−x )2+ (y −y )2≤r}
14.2 Limits and Continuity
Definition
An open disk with radius r centered at(x0,y0)∈R2 is the set
Br(x0,y0) ={(x,y)∈R2:p
(x−x0)2+ (y −y0)2<r}
A closed disk with radius r centered at(x0,y0)∈R2 is the set
B¯(x ,y ) ={(x,y)∈R2 :p
(x−x )2+ (y −y )2≤r}
14.2 Limits and Continuity
Definition
An open disk with radius r centered at(x0,y0)∈R2 is the set
Br(x0,y0) ={(x,y)∈R2:p
(x−x0)2+ (y −y0)2<r}
A closed disk with radius r centered at(x0,y0)∈R2 is the set
B¯(x ,y ) ={(x,y)∈R2 :p
(x−x )2+ (y −y )2≤r}
14.2 Limits and Continuity
Definition
An open disk with radius r centered at(x0,y0)∈R2 is the set
Br(x0,y0) ={(x,y)∈R2:p
(x−x0)2+ (y −y0)2<r}
A closed disk with radius r centered at(x0,y0)∈R2 is the set
B¯(x ,y ) ={(x,y)∈R2 :p
(x−x )2+ (y −y )2≤r}
(1)
Definition
The limit of f(x,y)is(x,y)approaches (x0,y0)is
(x,y)lim→(x0,y0)f(x,y),
which is the number L such that ∀ǫ >0, ∃δ >0 such that
|f(x,y)−L|< ǫ,whenever(x,y)∈Bδ(x0,y0)− {(x0,y0)}
(1)
Definition
The limit of f(x,y)is(x,y)approaches (x0,y0)is
(x,y)lim→(x0,y0)f(x,y),
which is the number L such that ∀ǫ >0, ∃δ >0 such that
|f(x,y)−L|< ǫ,whenever(x,y)∈Bδ(x0,y0)− {(x0,y0)}
(1)
Definition
The limit of f(x,y)is(x,y)approaches (x0,y0)is
(x,y)lim→(x0,y0)f(x,y),
which is the number L such that ∀ǫ >0, ∃δ >0 such that
|f(x,y)−L|< ǫ,whenever(x,y)∈Bδ(x0,y0)− {(x0,y0)}
(1)
Definition
The limit of f(x,y)is(x,y)approaches (x0,y0)is
(x,y)lim→(x0,y0)f(x,y),
which is the number L such that ∀ǫ >0, ∃δ >0 such that
|f(x,y)−L|< ǫ,whenever(x,y)∈Bδ(x0,y0)− {(x0,y0)}
(1)
Definition
The limit of f(x,y)is(x,y)approaches (x0,y0)is
(x,y)lim→(x0,y0)f(x,y),
which is the number L such that ∀ǫ >0, ∃δ >0 such that
|f(x,y)−L|< ǫ,whenever(x,y)∈Bδ(x0,y0)− {(x0,y0)}
Example:
1 f(x,y) = x2+y2 Find lim
(x,y)→(0,0)f(x,y) (ǫ, δargument)
2 lim
(x,y)→(0,0)
x2−2y2
x2+y2 (x =0,y =0)
3 lim
(x,y)→(0,0)
3xy
x2+y3 (y =mx)
Example:
1 f(x,y) = x2+y2 Find lim
(x,y)→(0,0)f(x,y) (ǫ, δargument)
2 lim
(x,y)→(0,0)
x2−2y2
x2+y2 (x =0,y =0)
3 lim
(x,y)→(0,0)
3xy
x2+y3 (y =mx)
Example:
1 f(x,y) = x2+y2 Find lim
(x,y)→(0,0)f(x,y) (ǫ, δargument)
2 lim
(x,y)→(0,0)
x2−2y2
x2+y2 (x =0,y =0)
3 lim
(x,y)→(0,0)
3xy
x2+y3 (y =mx)
Example:
1 f(x,y) = x2+y2 Find lim
(x,y)→(0,0)f(x,y) (ǫ, δargument)
2 lim
(x,y)→(0,0)
x2−2y2
x2+y2 (x =0,y =0)
3 lim
(x,y)→(0,0)
3xy
x2+y3 (y =mx)
(2) Remark:
Suppose that lim
(x,y)→(x0,y0)f(x,y) =L. If(xn,yn)n−→→∞(x0,y0),
then lim
n→∞
f(xn,yn) =L. 2
(2) Remark:
Suppose that lim
(x,y)→(x0,y0)f(x,y) =L. If(xn,yn)n−→→∞(x0,y0),
then lim
n→∞
f(xn,yn) =L. 2
(2) Remark:
Suppose that lim
(x,y)→(x0,y0)f(x,y) =L. If(xn,yn)n−→→∞(x0,y0),
then lim
n→∞
f(xn,yn) =L. 2
(2)Continuity
Definition
A function f(x,y)is continuous at(x0,y0)if 1 f(x,y)is defined at(x0,y0)
2 lim
(x,y)→(x0,y0)f(x,y)exists 3 lim
(x,y)→(x0,y0)f(x,y) = f(x0,y0)
(2)Continuity
Definition
A function f(x,y)is continuous at(x0,y0)if 1 f(x,y)is defined at(x0,y0)
2 lim
(x,y)→(x0,y0)f(x,y)exists 3 lim
(x,y)→(x0,y0)f(x,y) = f(x0,y0)
(2)Continuity
Definition
A function f(x,y)is continuous at(x0,y0)if 1 f(x,y)is defined at(x0,y0)
2 lim
(x,y)→(x0,y0)f(x,y)exists 3 lim
(x,y)→(x0,y0)f(x,y) = f(x0,y0)
(2)Continuity
Definition
A function f(x,y)is continuous at(x0,y0)if 1 f(x,y)is defined at(x0,y0)
2 lim
(x,y)→(x0,y0)f(x,y)exists 3 lim
(x,y)→(x0,y0)f(x,y) = f(x0,y0)
(2)Continuity
Definition
A function f(x,y)is continuous at(x0,y0)if 1 f(x,y)is defined at(x0,y0)
2 lim
(x,y)→(x0,y0)f(x,y)exists 3 lim
(x,y)→(x0,y0)f(x,y) = f(x0,y0)
Example 1:
Use the definition of continuity to show that f(x,y) =p
9+x2+y2 is continuous at (0,0)
Example 1:
Use the definition of continuity to show that f(x,y) =p
9+x2+y2 is continuous at (0,0)
Example 1:
Use the definition of continuity to show that f(x,y) =p
9+x2+y2 is continuous at (0,0)
Example 1:
Use the definition of continuity to show that f(x,y) =p
9+x2+y2 is continuous at (0,0)
Example 2:
Show that
f(x,y) =
4xy
x2+y2 , (x,y)6= (0,0) 0 , (x,y) = (0,0) is discontinuous at(0,0)
Example 2:
Show that
f(x,y) =
4xy
x2+y2 , (x,y)6= (0,0) 0 , (x,y) = (0,0) is discontinuous at(0,0)
Example 2:
Show that
f(x,y) =
4xy
x2+y2 , (x,y)6= (0,0) 0 , (x,y) = (0,0) is discontinuous at(0,0)
Example 2:
Show that
f(x,y) =
4xy
x2+y2 , (x,y)6= (0,0) 0 , (x,y) = (0,0) is discontinuous at(0,0)
