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
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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.