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L’Hopital’s Rule

L’Hopital’s Rule. Objective: To use L’Hopital’s Rule to solve problems involving limits. L’Hopital’s Rule. Forms of Type 0/0. We looked at several limits in chapter 2 that had the form 0/0. Some examples were:. Forms of Type 0/0.

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L’Hopital’s Rule

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  1. L’Hopital’s Rule Objective: To use L’Hopital’s Rule to solve problems involving limits.

  2. L’Hopital’s Rule

  3. Forms of Type 0/0 • We looked at several limits in chapter 2 that had the form 0/0. Some examples were:

  4. Forms of Type 0/0 • We looked at several limits in chapter 2 that had the form 0/0. Some examples were: • The first limit was solved algebraically and the next two were solved using geometric methods. There are other types that cannot be solved by either of these methods. We need to develop a more general method to solve such limits.

  5. Forms of Type 0/0 • Lets look at a general example.

  6. Forms of Type 0/0 • Lets look at a general example. • We will assume that f / and g / are continuous at x = a and g / is not zero. Since f and g can be closely approximated by their local linear approximation near a, we can say

  7. Forms of Type 0/0 • Lets look at a general example. • We know that: • So we can now say

  8. Forms of Type 0/0 • Lets look at a general example. • This will now become • This is know as L’Hopital’s Rule.

  9. Theorem 4.4.1 • L’Hopital’s Rule: Suppose that f and g are differentiable functions on an open interval containing x = a, except possibly at x = a, and that and • If exists or if this limit is then

  10. Example 1 • Find the following limit.

  11. Example 1 • Find the following limit. • In chapter 2 we would factor, cancel, and evaluate.

  12. Example 1 • Find the following limit. • Now, we can use L’Hopital’s Rule to solve this limit.

  13. Example 2 • Find the following limit.

  14. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0.

  15. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0. • Next, we use L’Hopital’s Rule to find the limit.

  16. Example 2 • Find the following limit.

  17. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0.

  18. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0. • Next, we use L’Hopital’s Rule to find the limit.

  19. Example 2 • Find the following limit.

  20. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0.

  21. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0. • Next, we use L’Hopital’s Rule to find the limit. • This is an asymptote. Do sign analysis to find the answer. ___+___|___+___ 0

  22. Example 2 • Find the following limit.

  23. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0.

  24. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0. • Next we use L’Hopital’s Rule to find the answer. • Again, sign analysis. ___-___|___+___ 0

  25. Example 2 • Find the following limit.

  26. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0.

  27. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0. • Next we use L’Hopital’s Rule to find the answer. • We apply L’Hopital’s Rule again.

  28. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0. • Next we use L’Hopital’s Rule to find the answer. • We apply L’Hopital’s Rule again.

  29. Example 2 • Find the following limit.

  30. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0.

  31. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0. • Next we use L’Hopital’s Rule to find the answer.

  32. Example 2 • Find the following limit. • First, we confirm that this of the form 0/0. • Next we use L’Hopital’s Rule to find the answer.

  33. Forms of Type • Theorem 4.4.2: Suppose that f and g are differentiable functions on an open interval containing x = a, except possibly at x = a, and that and • If exists or if this limit is then

  34. Example 3 • Find the following limits.

  35. Example 3 • Find the following limits. • First, we confirm the form of the equation.

  36. Example 3 • Find the following limits. • First, we confirm the form of the equation. • Next, apply L’Hopital’s Rule.

  37. Example 3 • Find the following limits.

  38. Example 3 • Find the following limits. • First, we confirm the form of the equation.

  39. Example 3 • Find the following limits. • First, we confirm the form of the equation. • Next, apply L’Hopital’s Rule. • This is the same form as the original, and repeated applications of L’Hopital’s Rule will yield powers of 1/x in the numerator and expressions involving cscx and cotx in the denominator being of no help. We must try another method.

  40. Example 3 • Find the following limits. • We will rewrite the last equation as

  41. Examine Exponential Growth • Look at the following graphs. What these graphs show us is that ex grows more rapidly than any integer power of x.

  42. Forms of Type • There are several other types of limits that we will encounter. When we look at the form we can make it look like 0/0 or and use L’Hopital’s Rule.

  43. Example 4 • Evaluate the following limit.

  44. Example 4 • Evaluate the following limit. • As written, this is of the form

  45. Example 4 • Evaluate the following limit. • As written, this is of the form • Lets rewrite this equation as

  46. Example 4 • Evaluate the following limit. • As written, this is of the form • Lets rewrite this equation as • Applying L’Hopital’s Rule

  47. Example 4 • Evaluate the following limit.

  48. Example 4 • Evaluate the following limit. • As written, this is of the form

  49. Example 4 • Evaluate the following limit. • As written, this is of the form • Lets rewrite this equation as

  50. Example 4 • Evaluate the following limit. • As written, this is of the form • Lets rewrite this equation as • Applying L’Hopital’s Rule

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