continuity_trick

continuity_trick - Passing the limit through a continuous...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
Passing the limit through a continuous function Math 8 2009, Chris Lyons Here’s a helpful trick that allows you to “pass” a limit from outside a continuous function to the inside. Proposition 1. Suppose that (i) g ( y ) is continuous at y = L , (ii) lim x a f ( x ) = L . Then lim x a g ( f ( x ) ) = g ± lim x a f ( x ) ² . Before giving a proof, here’s the intuitive idea behind the statement: let y = f ( x ). Then as x a we have y = f ( x ) L , and as y L we have g ( y ) g ( L ). In other words: as x a we have g ( f ( x ) ) g ( lim x a f ( x ) ) . Now here’s a real proof, using the theorem that the composition of two continuous functions is again continuous. Proof. Let’s make a new function: F ( x ) = ³ f ( x ) if x 6 = a L if x = a Then we know that F ( x ) is continuous at x = a because, by assumption (ii), we have lim x a F ( x ) = lim x a f ( x ) = L = F ( a ) . So now by Theorem 3.5 in the book, since
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 07/19/2010 for the course MA 8 taught by Professor Vuletic,m during the Fall '08 term at Caltech.

Ask a homework question - tutors are online