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

What is a limit ? When does a limit exist? Continuity Discontinuity Types of discontinuity. f (x) = x². When does a limit exist?. When does a limit exist?. When does a limit exist?. When does a limit exist?. +. In order to have a limit at a point At that point:

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

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  1. What is a limit ?When does a limit exist? ContinuityDiscontinuityTypes of discontinuity

  2. f (x) = x² When does a limit exist?

  3. When does a limit exist?

  4. When does a limit exist?

  5. When does a limit exist?

  6. + In order to have a limit at a point At that point: Left limit MUST EQUAL Right limit Lim f (x)= 1 Lim f (x)= 1 X  4ˉX  4 Lim f (x)= 1 X  4

  7. Limit exist Lim f (x)= 1 X  4

  8. Infinite Limits( are not limits)

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

  10. Substitution

  11. Factoring

  12. Factoring

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

  14. The Conjugate Method

  15. Continuityf is continuous at a if:

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

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

  18. 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: • no breaks in the graph • no holes • no jumps Discontinuity

  19. Everywhere Continuous Function 4 2 * 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.

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

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

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

  23. Removable discontinuity: thelimit of f exists at the point but f is not equal to the value of f at that point. 4 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,∞)

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

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