This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Lecture 3 : Friday April 3rd [email protected] Key concepts : Derivative, Differentiability Know how to find derivative from definition and using usual methods 3.1 Further examples of nonexistent limits Example 1. Prove that the limit as ( x,y ) → 0 of f ( x,y ) = ( x + y + x 2 ) / ( x y ) does not exist. Solution. Well let’s put y = mx to get f ( x,mx ) = (1 + m ) / (1 m ) + x/ (1 m ) for m 6 = 1. As x → 0, this is (1 + m ) / (1 m ) which depends on m . For example if m = 0 we get 1 whereas if m = 1 we get 0. Therefore the original limit does not exist. The next example is important because it shows that trying lines y = mx to show that a limit as ( x,y ) → (0 , 0) does not exist might fail. The point is, even if all limits lim x → f ( x,mx ) are equal (regardless of the value of m ) this does not mean lim ( x,y ) → (0 , 0) f ( x,y ) exists. Instead, we try a different curve through (0 , 0), in the next example, the curve we try is y = m √ x for different values of m . Example 2. Prove that the limit of xy 2 / ( x 2 + y 4 ) as ( x,y ) → 0 does not exist. Solution. If we put y = mx we get mx 3 / ( x 2 + m 4 x 4 ) = mx/ (1 + m 4 x 2 ). Now this limit as x → 0 exists, so we can’t conclude that the original limit does not exist. Therefore we0 exists, so we can’t conclude that the original limit does not exist....
View
Full
Document
This note was uploaded on 02/12/2010 for the course MATH 20E taught by Professor Enright during the Spring '07 term at UCSD.
 Spring '07
 Enright
 Math, Derivative, Limits

Click to edit the document details