Limmits

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Limmits

  1. 1. The number Lis called the limit of the function f(x) as x approaches a, written
  2. 2. x x
  3. 3. x 0 x
  4. 4. lim h x 2 x 1 lim h x 1 x 1 lim h x does not exist x 1
  5. 5. ) If f(x) = c ( a constant function), then for any a x a n n ) where n is a positive x a integer
  6. 6. ) If f x and gx x a x a exist , then x a x a x a
  7. 7. ) If lim f ( x ) and g x x a x a exist , then x a x a x a
  8. 8. - If lim f ( x ) exists, then for any x a constant k, lim k f ( x )] k lim f ( x ) x a x a
  9. 9. lim f ( x ) gx x a x a f (x ) lim f ( x ) x a lim x a g (x ) lim g ( x ) x a
  10. 10. ) If lim f ( x ) exits, and n is a positive x a integer n n x a x a
  11. 11. If does not exist, since
  12. 12. ) 1 lim p 0 where p > 0 x x 1 lim p 0 where p > 0 x x
  13. 13. The limit does not exist
  14. 14. If f(x) is a rational function and is the term with greatest power in the numerator and is the term with greatest power in the denominator, then
  15. 15. A function f(x) is continuous at a point b if and only if: ) f(x) is defined at x = b ) lim f(x) exist x b ) ) lim f(x) = f(b) x b
  16. 16. Show that f(x) = / (x- ) is continuous at x= and discontinuous at x = f( )= / lim x f(x) = / =f( ) At x= the function is not defined.
  17. 17. The function is not defined at x=- So it is not continuous at x=- , but the limit exist.
  18. 18. A polynomial function is continuous at all points of its domain.
  19. 19. Find all points of discontinuity of f(x) = x - x + X + X- X + X - = (x+ ) (x- ). The denominator is zero when x=- or x= x=- Thus the function is discontinuous at x=- and x= only.

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