1 / 23

SECTION 5.8

SECTION 5.8. NDETERMINATE FORMS AND L ’ HOSPITAL ’ S RULE. L ’ HOSPITAL ’ S RULE. Suppose f and g are differentiable and g ’ ( x ) ≠ 0 on an open interval I that contains a (except possibly at a ). Suppose or that In other words, we have an indeterminate form of type or ∞/∞.

naif
Download Presentation

SECTION 5.8

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. SECTION 5.8 NDETERMINATE FORMS AND L’HOSPITAL’S RULE

  2. L’HOSPITAL’S RULE • Suppose f and g are differentiable and g’(x) ≠ 0 on an open interval I that contains a (except possibly at a). Supposeor that • In other words, we have an indeterminate form of type or ∞/∞. • Then,if the limit on the right exists (or is ∞ or – ∞). 5.8

  3. Example 1 • Find • SOLUTION • Thus, we can apply l’Hospital’s Rule: 5.8

  4. Example 2 • Calculate • SOLUTION • We have • So, l’Hospital’s Rule gives: 5.8

  5. Example 2 SOLUTION • As ex→ ∞ and 2x → ∞ as x → ∞, the limit on the right side is also indeterminate. • However, a second application of l’Hospital’s Rule gives: 5.8

  6. Example 3 • Calculate • SOLUTION • As ln x→ ∞ and as x → ∞, l’Hospital’s Rule applies: • Notice that the limit on the right side is now indeterminate of type . 5.8

  7. Example 3 SOLUTION • However, instead of applying the rule a second time as we did in Example 2, we simplify the expression and see that a second application is unnecessary: 5.8

  8. Example 4 • Find • SOLUTION • Noting that both tan x–x→ 0 and x3 → 0 as x → 0, we use l’Hospital’s Rule: 5.8

  9. Example 4 SOLUTION • As the limit on the right side is still indeterminate of type , we apply the rule again: • Since , we simplify the calculation by writing: 5.8

  10. Example 4 SOLUTION • We can evaluate this last limit either by using l’Hospital’s Rule a third time or by writing tan x as (sin x)/(cos x) and making use of our knowledge of trigonometric limits. 5.8

  11. Example 4 SOLUTION • Putting together all the steps, we get: 5.8

  12. Example 5 • Find • SOLUTION • If we blindly attempted to use l-Hospital’s rule, we would get: 5.8

  13. Example 5 SOLUTION • This is wrong. • Although the numerator sin x→ 0 as x → p–, notice that the denominator (1 – cos x) does not approach 0. • So, the rule can’t be applied here. • The required limit is, in fact, easy to find because the function is continuous at p and the denominator is nonzero there: 5.8

  14. Example 6 • Evaluate • SOLUTION • The given limit is indeterminate because, as x→ 0+, the first factor (x) approaches 0, whereas the second factor (ln x) approaches – ∞. 5.8

  15. Example 6 SOLUTION • Writing x = 1/(1/x), we have 1/x→ ∞ as x → 0+. • So, l’Hospital’s Rule gives: 5.8

  16. Example 7 • Compute • SOLUTION • First, notice that sec x → ∞ and tan x → ∞ as x → (p/2)–. • So, the limit is indeterminate. 5.8

  17. Example 7 SOLUTION • Here, we use a common denominator: • Note that the use of l’Hospital’s Rule is justified because 1 – sin x→ 0 and cos x → 0 as x → (p/2)–. 5.8

  18. INDETERMINATE POWERS • Several indeterminate forms arise from the limit 5.8

  19. Example 8 • Calculate • SOLUTION • First, notice that, as x→ 0+, we have 1 + sin 4x → 1 and cot x → ∞. • So, the given limit is indeterminate. • Let y = (1 + sin 4x)cot x • Then, ln y = ln[(1 + sin 4x)cot x] = cot x ln(1 + sin 4x) 5.8

  20. Example 8 SOLUTION • So, l’Hospital’s Rule gives: 5.8

  21. Example 8 SOLUTION • So far, we have computed the limit of ln y. • However,what we want is the limit of y. • To find this, we use the fact that y =eln y: 5.8

  22. Example 9 • Find • SOLUTION • Notice that this limit is indeterminate since 0x = 0 for any x > 0 but x0 = 1 for any x ≠ 0. 5.8

  23. Example 9 SOLUTION • We could proceed as in Example 8 or by writing the function as an exponential: xx = (eln x)x = ex ln x • In Example 6, we used l’Hospital’s Rule to show that • Therefore, 5.8

More Related