1. What is a limit ? When does a limit exist? Continuity Discontinuity Types of discontinuity

2. What is a limit? A limit of a function is an intended “height” of a function at a point Lim f (x) = 4 X  2

3. When does a limit exist?

4. f (x) = x²

5. When does a limit exist?

6. With a plus sign

7. When does a limit exist? With a minus sign

8. In order to have a limit at a point At that point: Left limit MUST EQUAL Right limit Lim f (x)= 1 X  4ˉ X  4 Lim f (x)= 1 X  4 With no sign +

9. Limit exist Lim f (x)= 1 X  4

11. Infinite Limits ( are not limits)

12. How to evaluate a limit, in case of algebraic functions? 1.Finding the value of the function at the point (Substitution in the formula of the function) If the function is continuous at the point 2.Factoring (The case 0/0 ) 3.The Conjugate Method (The case 0/0 )

13. Substitution

14. Factoring

16. The Conjugate Method What is a conjugate? X-16 = (√x - 4) (√x + 4) AND (√x + 4) is the conjugate of (√x - 4) respect to X-16 (√x - 4) is the conjugate of (√x + 4) respect to X-16

17. The Conjugate Method

18. Continuity f is continuous at a if:

19. Example of a function f which is discontinuous, but continuous from the right. lim f (x) = 4 = f (2) X  2+

20. Example of a function f which is discontinuous, but continuous from the left. lim f (x) = 4 = f (2) X  2ˉ

21. Discontinuity A function is discontinues at a if the limit at a is not equal to the value f (a) A continuous function on R should have: 1.no breaks in the graph 2.no holes 3.no jumps

22. Everywhere Continuous Function 2 4 * Since f(2) is also equal to 4; then = f(2) The function is continuous at 2. And since it’s also continuous at all other point’s in R; then it’s everywhere continuous.

23. Type of discontinuity 1. Removable discontinuity 2. infinite discontinuity 3. jump discontinuity

24. Jump discontinuity 1 7 4 f is continuous on the intervals (-∞,1 ) and [1, ∞ )

25. Infinite discontinuity 3 4 The function is continuous on the interval (-∞, 3] and (3, ∞ ) Notice that the function is continuous from the left at 3

26. Removable discontinuity : the limit of f exists at the point but f is not equal to the value of f at that point. 3 Notice that the function is neither continues from the left nor continuous from the right at 3 The function is continuous on the interval (-∞,3) and (3,∞) 4

27. Let 1 2 3 -2 (1,3) f is not defined at 1 but exists and equal to 3 Removable discontinuity: the limit of f exists at the point but f is not defined at that point.