12.2Limits - 12. R2 lim f(x = L ,then xc f(x)isclosetoL Def...

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12.2 Limits and Continuity R 2 : lim x c f ( x ) = L means that if x is “close” to c, then f(x) is “close” to L. Def: lim x c f ( x ) = L ↔ ∀ ε > 0, δ > 0 such that if x c < δ then f ( x ) L < ε R 3 : lim ( x , y ) ( a , b ) 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
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Ex. lim x 3 2 x 3 = Ex. lim x 3 x 2 9 x 3 = Ex. lim x 3 1 x 3 = Ex. lim x , y ( ) 0,1 ( ) x 2 y xy + 3 = Ex. lim x , y ( ) 0,0 ( ) x 2 2 xy + 5 y 2 3 x 2 + 4 y 2 =
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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 3 x 2 + 4 y 2 = y‐axis (x = 0) y = kx
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Ex. lim x , y ( ) 0,0 ( ) 2 x 2 y x 4 + y 2 = Ex. lim x , y ( ) 0,0 ( ) x y x y =
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Ex. lim x , y ( ) 0,0 ( ) xy x 2 + y 2 =
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Polar Coordinates x = r cos θ y = r sin θ Let r approach 0 Ex. lim x , y ( ) 0,0 ( ) cos x 3 y 4 x 2 + y 2
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