math237_a2 - SOLUTIONS FOR ASSIGNMENT 2 A3 Consider f R 2...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: SOLUTIONS FOR ASSIGNMENT 2 A3. Consider f : R 2 → R defined by f ( x,y ) = sin( xy ) ln( x 2 + y 2 + 1) , ( x,y ) negationslash = (0 , 0) f (0 , 0) = . Prove that f x (0 , 0) and f y (0 , 0) exist, but that f is not continuous at (0 , 0) . From the definition of first order partial derivatives we have: f x (0 , 0) = lim h → f (0 + h, 0) − f (0 , 0) h = lim h → sin 0 ln( h 2 +1) − h = 0 , and, similarly, f y (0 , 0) = lim h → f (0 , 0 + h ) − f (0 , 0) h = lim h → sin 0 ln( h 2 +1) h = 0 . Since the above limits exist we conclude that f x (0 , 0) and f y (0 , 0) exist. In order to show that f is not continuous at (0 , 0) we need to show that lim ( x,y ) → (0 , 0) sin( xy ) ln( x 2 + y 2 + 1) negationslash = 0 . We evaluate this limit along the straight line paths y = mx, ∀ m ∈ R . We have: lim x → sin( mx 2 ) ln((1 + m 2 ) x 2 + 1) = m 1 + m 2 lim x → ((1 + m 2 ) x 2 + 1) cos ( mx 2 ) (by l ′ Hospital ′ s rule) = m 1 + m 2 , ∀ m ∈ R . Since the above limit depends on the slope m of the straight line paths, it follows from the two paths’ rule that the limit lim ( x,y ) → (0 , 0) sin( xy ) ln( x 2 + y 2 + 1) does not exist and hence it cannot equal 0 = f (0 , 0) . Thus f is not continuous at (0 , 0) and this 1 ends the proof. B1 (v). For the function f ( x,y ) = x 2 − 6 y 2 | x | + 3 | y | determine whether or not lim ( x,y ) → (0 , 0) f ( x,y ) exists and, if possible, define f (0 , 0) so as to make the...
View Full Document

This note was uploaded on 07/28/2009 for the course MATH MATH137 taught by Professor Oancea during the Spring '08 term at Waterloo.

Page1 / 5

math237_a2 - SOLUTIONS FOR ASSIGNMENT 2 A3 Consider f R 2...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online