Lecture 12-2

# Lecture 12-2 - easy polar conversions i.e x 2 y 2 that...

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12.2 Limits and Continuity in Higher Dimen- sions Limits In R 2 : lim x c f ( x ) = L lim x c f ( x ) = L ⇐⇒ In R 3 : Defn. Limit lim ( x,y ) ( a,b ) f ( x, y ) = L ⇐⇒ 1

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In other words, lim ( x,y ) ( a,b ) f ( x, y ) = L means Examples in R 2 1. lim x 3 2 x - 3 2. lim x 3 x 2 - 9 x - 3 3. lim x 2 1 x - 2 Example 1 lim ( x,y ) (0 , 1) x 2 - y xy + 3 Example 2 lim ( x,y ) (0 , 0) x 2 - 2 xy + 5 y 2 3 x 2 + 4 y 2 2
For a limit to exist, In R 2 : In R 3 : Back to Example 2 Let’s investigate the limit along different paths. lim ( x,y ) (0 , 0) x 2 - 2 xy + 5 y 2 3 x 2 + 4 y 2 Note: The path you select MUST go through the point that the limit is approaching!!!! 3

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Example 3 lim ( x,y ) (0 , 0) 2 x 2 y x 4 + y 2 Example 4 lim ( x,y ) (0 , 0) x - y x - y 4
Polar Coordinates Recall: x = r cos θ x 2 + y 2 = r 2 y = r sin θ tan θ = y x Let r 0. Use polar coordinates to help evaluate limits approaching (0,0).

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Unformatted text preview: easy polar conversions, i.e. x 2 + y 2 , that cause cancelations in your function. Example 1 lim ( x,y ) → (0 , 0) cos ± x 2-y 4 x 2 + y 2 ² Example 2 lim ( x,y ) → (0 , 0) xy p x 2 + y 2 5 Continuity Defn. Continuous f ( x, y ) is continuous at ( a, b ) if 1. 2. 3. Examples Find where the following functions are continuous. 1. f ( x, y ) = x-y x + y 2. f ( x, y, z ) = √ x-y + z 3. f ( x, y ) = 2 6 Do. 1. lim ( x,y,z ) → (1 , 3 ,-2) x + 2 y-z xyz 2. lim ( x,y ) → (0 , 0) x 2-y x 2 + y 7...
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