5 2 limit-ofa_function

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5 2 limit-ofa_function

  1. 1. Limit of a Function
  2. 2. <ul><li>The concept of a limit is central to the study of calculus. </li></ul><ul><li>Differentiation and integration are each based on the notion of a limit. </li></ul><ul><li>Limit of f ( x ) as x approaches a – the value that f ( x ) approaches as x gets closer to a </li></ul><ul><li>In symbols: </li></ul>
  3. 3. <ul><li>Observe the values of f ( x ) = x 2 + 1 as x approaches 3: </li></ul><ul><li>We say that . </li></ul>x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4 f ( x ) 5 7.25 9.41 9.94 9.994 10 10.006 10.06 10.61 13.25 17
  4. 5. <ul><li>Given . Find the limit of f ( x ) as x approaches 3. . </li></ul><ul><li>We still say that . </li></ul>x 2 2.5 2.9 2.99 2.999 3 3.001 3.01 3.1 3.5 4 f ( x ) 5 7.25 9.41 9.94 9.994 15 10.006 10.06 10.61 13.25 17
  5. 6. <ul><li>Given . Find the limit of f ( x ) as x approaches 2. . </li></ul>x 1 1.5 1.9 1.99 1.999 2 2.001 2.01 2.1 2.5 3 f ( x ) 3 4 4.8 4.98 4.998 4 4.004 4.04 4.41 6.25 9
  6. 7. <ul><li>The limit of the function f exists only if the left-hand and right-hand limits both exist and are equal. </li></ul><ul><li>exists if and only if </li></ul>
  7. 8. Limit Theorems
  8. 9. <ul><li>L1. The limit of a constant function is the constant. </li></ul><ul><li>Examples. </li></ul>
  9. 10. <ul><li>L2. If f is a polynomial function, then </li></ul><ul><li>Example. </li></ul>
  10. 11. <ul><li>L3. The limit of a sum is the sum of limits. </li></ul><ul><li>Example. </li></ul>
  11. 12. <ul><li>L4. The limit of a constant times a function is the constant times the limit of the function. </li></ul><ul><li>Example. </li></ul>
  12. 13. <ul><li>L5. The limit of a product is the product of the limits. </li></ul><ul><li>Example. </li></ul>
  13. 14. <ul><li>L6. The limit of a quotient is the quotient of the limits. </li></ul><ul><li>Example. </li></ul>
  14. 15. <ul><li>L7. The limit of a function raised to an exponent is the limit of the function raised to that exponent. </li></ul><ul><li>Example. </li></ul>
  15. 16. <ul><li>Example. Find . </li></ul>
  16. 17. <ul><li>Example. Find . </li></ul>
  17. 18. <ul><li>Example. Given Find . . </li></ul>
  18. 19. <ul><li>1.a Find . </li></ul>
  19. 20. <ul><li>1.d Find . </li></ul>
  20. 21. <ul><li>2.c Find . </li></ul>
  21. 22. <ul><li>3.a Find . </li></ul>
  22. 23. <ul><li>3.c Find . </li></ul>
  23. 24. <ul><li>4.b Find . </li></ul>
  24. 25. <ul><li>5.a Suppose and . Find . . </li></ul>
  25. 26. <ul><li>7. Given . Find . . </li></ul>

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