12.2Limits - 12.2
Limits
and
Continuity
 R2:
 lim...

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Unformatted text preview: 12.2
Limits
and
Continuity
 R2:
 lim f (x) = L 
means
that
if
x
is
“close”
to
c,
then
 x→c f(x)
is
“close”
to
L.
 
 Def:
 lim f (x) = L ↔ ∀ε > 0, ∃δ > 0 such that x→c 
 if x − c < δ then f (x) − L < ε 
 
 
 R3:

 (x,y )→(a,b ) 
 lim f (x, y) = L ↔ ∀ε > 0, ∃δ > 0 such that if 0 < 
 
 
 
 
 
 ( x − a) 2 + ( y − b) 2 < δ then f (x, y) − L < ε 
 Ex.
 lim x→3 
 
 2x − 3 = 
 x2 − 9 lim =
 Ex.
 x→3 x − 3 
 
 
 1 lim = Ex.
 x→3 x − 3 
 
 
 Ex.
 ( x,y )→( 0,1) 
 
 lim x2 − y = xy + 3 
 Ex.
 ( x,y )→( 0,0) 
 lim x 2 − 2xy + 5y 2 = 2 2 
 3x + 4 y For
a
limit
to
exist,
it
must
be
the
same
along
every
approach
 path.

Let
the
function
approach
along
the
x‐axis
(y
=
0)
 ( x,y )→( 0,0) 
 
 
 
 lim x 2 − 2xy + 5y 2 = 2 2 
 3x + 4 y y‐axis
(x
=
0)
 
 
 
 
 
 y
=
kx
 
 
 
 
 
 
 Ex.
 ( x,y )→( 0,0) 
 
 
 
 
 
 
 
 
 
 Ex.
 ( x,y )→( 0,0) 
 
 
 
 
 
 lim 2x 2 y 4 2 =
 x +y lim x−y = x− y 
 Ex.
 ( x,y )→( 0,0) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 lim xy x +y 2 2 = 
 Polar
Coordinates
 x = r cosθ Let
r
approach
0
 
 y = r sin θ 
 Ex.
 ( x,y )→( 0,0) 
 
 
 
 
 
 
 
 
 
 
 
 
 
 lim ⎛ x3 − y4 ⎞ cos⎜ 2 2⎟ ⎝x + y ⎠
 Do:

Find
the
following
limits,
if
they
exist.
 1.
 ( x,y,z )→(1,3,−2) 
 lim x + 2y − z 
 xyz 2.
 ( x,y )→( 0,0) 
 lim x2 − y x2 + y 
 x2 − y2 x+y 
 3.
 ( x,y )→( 0,0) 
 
 
 
 
 
 
 
 
 
 
 lim Continuity
 Def:

f(x,
y)
is
continuous
at
(a,
b)
if:
 1.
 (x,ylim ) f (x, y) 
exists.
 )→(a,b 2.
f(a,
b)
exists.
 3.
 (x,ylim ) f (x, y) = f (a,b) 
 )→(a,b 
 
 Ex.

 
 
 
 Ex.

 
 
 
 Ex.

 
 x−y f (x, y) = x + y
 f (x, y,z) = x − y + z
 f (x, y) = 2 
 ...
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This note was uploaded on 03/05/2009 for the course MATH 2224 taught by Professor Mecothren during the Spring '03 term at Virginia Tech.

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