12.2 Limits

# 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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