**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 *lim _{x→a}f(x)=L* if no matter how close (ε) you want

- Set the top slider (adjust it and click the button below it) to
*1.4*. What, approximately, is*lim*?_{x→1.4}f(x) - Set the second slider to
*0.72*. Do you expect that*lim*? (Check with your answer to question 1.)_{x→1.4}f(x)=0.72 - Set the third slider to
*0.3*. What does the third slider control? - 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? - Does your answer to question 4 imply that
*lim*? Why or why not?_{x→a}f(x)=0.72 - 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*. - 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*. - Interpret your answers to questions 6 & 7.
- 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?
- Reset, and set
*a=1.4*and*L=0.6*. Try different values of ε and δ. Is*lim*? Why or why not?_{x→1.4}f(x)=0.6 - Reset, and set
*a=-1*and*L=2*. Try different values of ε and δ. Is*lim*? Why or why not?_{x→-1}f(x)=2 - Reset, and set
*a=-1*and*L=-1*. Try different values of ε and δ. Is*lim*? Why or why not?_{x→-1}f(x)=-1 - Reset, and set
*a=2*and*L=-1*. Try different values of ε and δ. Is*lim*? Why or why not?_{x→2}f(x)=-1 - Reset, and set
*a=1*and*L=-1*. Try different values of ε and δ. Is*lim*? Why or why not?_{x→1}f(x)=-1 - Does
*lim*exist? Why or why not? How about_{x→1}f(x)*lim*? Why or why not?_{x→-2}f(x)