Lecture 5 limit laws
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This document discusses various limit laws and techniques for calculating limits, including: 1) Using limit laws and graphs to evaluate specific limits, such as the product law 2) Limits involving indeterminate forms like 0/0 cannot always be directly substituted and may require alternative techniques 3) Other techniques include factoring, rationalizing fractions, and applying laws for powers, roots, and other specific limits.
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![cont’d
5
Example 1(b) – Solution
We see that limx®1 f (x) = 2. But limx®1 g(x) does not exist
because the left and right limits are different:
So we can’t use Law 4 for the desired limit. But we can use
Law 4 for the one-sided limits:
The left and right limits aren’t equal, so limx®1 [f (x)g(x)] does
not exist.](https://image.slidesharecdn.com/lecture5-limitlaws-140916111716-phpapp01/85/Lecture-5-limit-laws-5-320.jpg)
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Lecture 5 limit laws
- 1. Calculating Limits Using 2.3 the Limit Laws
- 2. Calculating Limits Using the Limit Laws 2
- 3. 3 Example 1 Use the Limit Laws and the graphs of f and g in Figure 1 to evaluate the following limits, if they exist. Figure 1
- 4. 4 Example 1(a) – Solution From the graphs of f and g we see that and Therefore we have = 1 + 5(–1) = –4 (by Law 1) (by Law 3)
- 5. cont’d 5 Example 1(b) – Solution We see that limx®1 f (x) = 2. But limx®1 g(x) does not exist because the left and right limits are different: So we can’t use Law 4 for the desired limit. But we can use Law 4 for the one-sided limits: The left and right limits aren’t equal, so limx®1 [f (x)g(x)] does not exist.
- 6. 6 Example 1(c) – Solution The graphs show that and Because the limit of the denominator is 0, we can’t use Law 5. The given limit does not exist because the denominator approaches 0 while the numerator approaches a nonzero number. cont’d Figure 1
- 7. 7 Power Law If we use the Product Law repeatedly with g(x) = f (x), we obtain the following law. Power Law
- 8. 8 Powers & Roots
- 9. 9 Two Basic Limits
- 10. 10 Example
- 11. 11 Calculating Limits – Direct Substitution Functions with the Direct Substitution Property are called continuous at a.
- 12. 12 Calculating Limits – Finding Find a function that agrees everywhere but at the “problem point” When direct substitution yields 0/0.
- 13. 13 Finding Limits - Factoring
- 14. Finding Limits – Factoring - Example 14
- 15. 15 Finding Limits - Rationalization
- 17. 17 Finding Limits - Fractions
- 18. Finding Limits – Fractions - Example 18