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Problem 1 For all non-zero real numbers x + 0 and y + 0, define a function f by 1 f (x, y) = 1 O or y = 0. Formula (1) is not defined if x = This problem focuses on finding a value L so that f extends to a function that is continuous. (1.4) Determine a number L such that for each ɛ > 0 there is a d > 0 such that if 0 < |x| < 8 or 0 < ]y] < d, then |f(x, y) – L| < ɛ: Find L, which may not depend on e or (x, y): (1.5) For the number L in the preceding item, for each e > 0 find a d > 0 such that if 0 < |x| < d or 0 < [y] < 8, then |f (x, y) – L| < ɛ: Find d in terms of ɛ, only in terms of ɛ, so that 8 may not depend on (x, y):

Problem 1 For all non-zero real numbers x + 0 and y + 0, define a function f by 1 f (x, y) = 1 O or y = 0. Formula (1) is not defined if x = This problem focuses on finding a value L so that f extends to a function that is continuous. (1.4) Determine a number L such that for each ɛ > 0 there is a d > 0 such that if 0 < |x| < 8 or 0 < ]y] < d, then |f(x, y) – L| < ɛ: Find L, which may not depend on e or (x, y): (1.5) For the number L in the preceding item, for each e > 0 find a d > 0 such that if 0 < |x| < d or 0 < [y] < 8, then |f (x, y) – L| < ɛ: Find d in terms of ɛ, only in terms of ɛ, so that 8 may not depend on (x, y):

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section5.3: The Natural Exponential Function
Problem 54E
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Problem 1 For all non-zero real numbers x + 0 and y + 0, define a function f by 1 f (x, y) = 1 O or y = 0. Formula (1) is not defined if x = This problem focuses on finding a value L so that f extends to a function that is continuous. (1.4) Determine a number L such that for each ɛ > 0 there is a d > 0 such that if 0 < |x| < 8 or 0 < ]y] < d, then |f(x, y) – L| < ɛ: Find L, which may not depend on e or (x, y): (1.5) For the number L in the preceding item, for each e > 0 find a d > 0 such that if 0 < |x| < d or 0 < [y] < 8, then |f (x, y) – L| < ɛ: Find d in terms of ɛ, only in terms of ɛ, so that 8 may not depend on (x, y):
Transcribed Image Text:Problem 1 For all non-zero real numbers x + 0 and y + 0, define a function f by 1 f (x, y) = 1 O or y = 0. Formula (1) is not defined if x = This problem focuses on finding a value L so that f extends to a function that is continuous. (1.4) Determine a number L such that for each ɛ > 0 there is a d > 0 such that if 0 < |x| < 8 or 0 < ]y] < d, then |f(x, y) – L| < ɛ: Find L, which may not depend on e or (x, y): (1.5) For the number L in the preceding item, for each e > 0 find a d > 0 such that if 0 < |x| < d or 0 < [y] < 8, then |f (x, y) – L| < ɛ: Find d in terms of ɛ, only in terms of ɛ, so that 8 may not depend on (x, y):
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