Explanation Definition used: “A function f ( x , y ) is continuous at ( a , b ) if lim ( x , y ) → ( a , b ) f ( x , y ) = f ( a , b ) ”. Calculation: The function is, lim ( x , y ) → ( 0 , 0 ) e − x 2 − y 2 − 1 x 2 + y 2 . The polar coordinate of the point ( x , y ) is ( r , θ ) where r ≥ 0 . Note that r → 0 + as ( x , y ) → ( 0 , 0 ) . The value of x and y can be represented in polar coordinates as, x = r cos θ and y = r sin θ Since every rational function is continuous, the given rational function is continuous. So, substitute x = r cos θ and y = r sin θ directly in the given continuous function and obtain the required limit as shown below. lim ( x , y ) → ( 0 , 0 ) e − x 2 − y 2 − 1 x 2 + y 2 = lim r → 0 + e − r 2 cos 2

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Author: James Stewart

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Limits of Multivariable Functions

Multivariable calculus is an extension of single variable calculus. It involves several variables instead of just one. Assume there is an open set containing points (x0, y0), let f be a function defined in that open interval except for the points (x0, y0…

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Chapter 14.2, Problem 41E

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To find: The limit of the function lim(x,y)→(0,0)e−x2−y2−1x2+y2 , using polar coordinates.

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Chapter 14.2, Problem 40E

Chapter 14.2, Problem 42E

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The limit of the function lim ( x , y ) → ( 0 , 0 ) e − x 2 − y 2 − 1 x 2 + y 2 , using polar coordinates.

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To find: The limit of the function lim(x,y)→(0,0)e−x2−y2−1x2+y2 , using polar coordinates.

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Chapter 14 Solutions