Goal: To understand the technical (ε-δ) definition of the limit.


The blue curve represents the graph of a function y=f(x) which clearly is discontinuous at some points. We say that limx→af(x)=L if no matter how close (ε) you want f(x) to get to L, we can find some distance (δ) so that whenever x is within that distance of a (but not just a, then f(x) is within the desired distance of L. In the language of mathematics, we write

limx→af(x)=L iff ∀ ε>0 ∃ δ>0 ∋ (|x-a|<δ, x≠a) ⇒ |f(x)-L|<ε.
In the applet above, you first chose a value for a that you like by adjusting the slider to the desired value and then clicking on the button below the slider. Once the a value has been chosen, you next choose a value for L using an analogous method. Next, a value of ε is chosen, and finally, a value of δ can be searched for...though it may or may not exist.
  1. Set the top slider (adjust it and click the button below it) to 1.4. What, approximately, is limx→1.4f(x)?
  2. Set the second slider to 0.72. Do you expect that limx→1.4f(x)=0.72? (Check with your answer to question 1.)
  3. Set the third slider to 0.3. What does the third slider control?
  4. Can you find a value of δ so that |x-1.4|<ε|f(x)-0.72|<δ? If so, what is the largest such value? If not, why not?
  5. Does your answer to question 4 imply that limx→af(x)=0.72? Why or why not?
  6. Press "RESET." Again set the first slider on 1.4 and the second on 0.72, but set ε=0.2. Again find the largest δ so that |x-1.4|<ε|f(x)-0.72|<δ.
  7. Press "RESET." Again set the first slider on 1.4 and the second on 0.72, but set ε=0.1. Again find the largest δ so that |x-1.4|<ε|f(x)-0.72|<δ.
  8. Interpret your answers to questions 6 & 7.
  9. If the applet allowed smaller values of δ, would we be able to find one that works in question 7? Why or why not? What if it allowed smaller values of ε too?
  10. Reset, and set a=1.4 and L=0.6. Try different values of ε and δ. Is limx→1.4f(x)=0.6? Why or why not?
  11. Reset, and set a=-1 and L=2. Try different values of ε and δ. Is limx→-1f(x)=2? Why or why not?
  12. Reset, and set a=-1 and L=-1. Try different values of ε and δ. Is limx→-1f(x)=-1? Why or why not?
  13. Reset, and set a=2 and L=-1. Try different values of ε and δ. Is limx→2f(x)=-1? Why or why not?
  14. Reset, and set a=1 and L=-1. Try different values of ε and δ. Is limx→1f(x)=-1? Why or why not?
  15. Does limx→1f(x) exist? Why or why not? How about limx→-2f(x)? Why or why not?