12.2 Limits - 1 2 .2 
L im it s 
a n d 
C o n t in u...

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Unformatted text preview: 1 2 .2 
L im it s 
a n d 
C o n t in u it y 
 R2:
 lim f ( x ) = L 
means
that
if
x
is
“close”
to
c,
then
 x→ c f( x ) 
is 
“ c l o s e ” 
t o 
L .
 
 D e f :
 lim f ( x ) = L ↔ ∀ε > 0, ∃δ > 0 such that x→ c 
 if x − c < δ then f ( x ) − L < ε 
 
 
 
 R 3:
 
 lim ( x , y )→( a , b ) if 0 < 
 
 
 
 
 
 f ( x, y ) = L ↔ ∀ε > 0, ∃δ > 0 such that ( 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 
 
 
 lim x2 − y = xy + 3 
 lim x 2 − 2 xy + 5 y 2 = 2 2 
 3x + 4 y Ex.
 ( x ,y )→( 0,1) 
 
 Ex.
 ( x ,y )→( 0,0) 
 For
a
limit
to
exist,
it
must
be
the
same
along
every
approach
 path.

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

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

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

 x−y f ( x, y ) = x + y
 
 
 
 Ex.

 f ( x, y, z) = 
 
 
 Ex.

 
 f ( x, y ) = 2 
 x − y + z
 ...
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This note was uploaded on 10/02/2011 for the course AERO 1234 at Virginia Tech.

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