1. Unit 1: Limits Helga Hufflepuff

2. A limit is

3. A limit is the y-value of a graph as x approaches from both sides

6. A limit does not exist when the y-value is different as you approach the same x value from different sides

7. If the function is continuous… you can draw the function without ever having to pick up your pen from the graph The limit value is the same as the function value

8. Discontinuity Types of Discontinuity: Jump Hole Infinite (Asymptote)

9. Left- and Right-Hand Limits Sometimes, limits do not approach the same y-value as they approach the same x-value, but the limit still exists If approaching from the left - If approaching from the right +

10. L’Hopital’s Rule Used to calculate limits involving indeterminate forms; 0/0 or ∞/∞ Is there a limit?

11. 1) Let’s find the derivative of y/x: 2) Since x’=1, we can now find the limit. The limit is 1

12. The Squeeze Theorem If the limit as h(x) approaches a is equal to the limit as f(x) approaches a—at point L-- and f(x)<g(x)<h(x), then g(x) equals L

13. Continuous VS. Differentiable Differentiable– Does the derivative exist? If a function is not continuous it cannot be differentiable on all reals Continuous on all reals. Does the derivative exist

14. Differentiable? If the function is continuous, we only need to worry about where derivative is undefined Cusps Corners At x=2, the derivative is undefined The function is differentiable on all reals except for where x=2

15. Continuous and Differentiable

16. In the Absolute Value function, the derivative is undefined at x=0 NOT DIFFERENTIABLE

17. Intermediate Value Theorem If the function is continuous from [a,b], then there must be a point c in the interval [a,b] and it must have a y-value that is between f(a) and f(b)