Worksheet_4 - f takes on the constant value 1. 3. Consider...

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Discussion – Thursday, September 8th 1. For each of the following functions f : R 2 R , draw a sketch of the graph together with pictures of some level sets. (a) f ( x , y ) = xy (b) f ( x ) =| x | . Please note here that x is a vector. In coordinates, this function is f ( x , y ) = p x 2 + y 2 . For (a), the result is one of the many quadric surfaces. What is the name for this type? Is the graph in (b) also a quadric surface? 2. Consider the function f : R 2 R given by f ( x , y ) = 2 x 3 y x 6 + y 2 for ( x , y ) 6= 0 In this problem, you’ll consider lim ( x , y ) 0 f ( x , y ). (a) Look at the values of f on the x - and y -axes. What do these values show the limit lim ( x , y ) 0 f ( x , y ) must be if it exists ? (b) Show that along each line in R 2 through the origin, the limit of f exists and is 0. (c) Despite this, show that the limit lim ( x , y ) 0 f ( x , y ) does not exist by finding a curve over which
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Unformatted text preview: f takes on the constant value 1. 3. Consider the function f : R 2 R given by f ( x , y ) = xy 2 p x 2 + y 2 for ( x , y ) 6= In this problem, youll show lim h f ( h ) = 0. (a) For = 1/2, nd some > 0 so that when 0 <| h |< we have | f ( h ) |< . Hint: As with the example in class, the key is to relate | x | and | y | with | h | . (b) Repeat with = 1/10. (c) Now show that lim h f ( h ) = 0. That is, given an arbitrary > 0, nd a > 0 so that that when 0 <| h |< we have | f ( h ) |< . (d) Explain why the limit laws that you learned in class on Wednesday arent enough to compute this particular limit....
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