limits - NOTES ON LIMITS In these notes we give the...

Info icon This preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon
NOTES ON LIMITS In these notes we give the definition of the limit of a function ; the material on se- quences is in Appendix D.2 of the book. Our goal is to see how to use this definition to prove some of the types of statements which you will see in Math 104. We are not trying to fully explain what a limit actually is, and so we give no motivation nor further insight into the definition; you will see all of this in Math 104. Again, here we just want to see some basic examples of proofs involving ’s and δ ’s. Definition 1. Let f : R R be a function. We say that L R is the limit of f as x approaches a R if for every > 0, there exists δ > 0 such that | f ( x ) - L | < if 0 < | x - a | < δ . We denote this by writing lim x a f ( x ) = L . Proposition 1. Let f : R R be a constant function, f ( x ) = c . Then for any a R , lim x a f ( x ) = c Proof. Let a R . Let > 0 and let δ = 1. Then if 0 < | x - a | < δ , we have | f ( x ) - c | = | c - c | = 0 < . Hence lim x a f ( x ) = c . Proposition 2. Let f : R R be the function defined by f ( x ) = x + 3 . Then lim x 1 f ( x ) = 4 .
Image of page 1

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

View Full Document Right Arrow Icon
Image of page 2
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    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.

    Student Picture

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

  • Left Quote Icon

    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.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    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.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern