continuity_trick

continuity_trick - Passing the limit through a continuous...

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

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 ) . 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 ) = f ( x ) if x = 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 g ( y ) is continuous at y = L and F ( x ) is continuous at x = a , the composition g ( F ( x )) is continuous at x = a . But what does this mean? It means lim x a g ( F ( x )) = g ( F ( a )) . Therefore lim x a g ( f ( x )) = lim x a g ( F ( x )) = g ( F ( a )) = g ( L ) = g lim x a f ( x ) . Here’s a handy consequence of this result: Corollary 2. Let g be a function that is continuous everywhere. If
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern