1. Copyright © 2011 Pearson Education, Inc. Slide 12.3-1 12.3 One-Sided Limits Limits of the form are called two-sided limits since the values of x get close to a from both the right and left sides of a. Limits which consider values of x on only one side of a are called one-sided limits.

2. Copyright © 2011 Pearson Education, Inc. Slide 12.3-2 12.3 One-Sided Limits The right-hand limit, is read “the limit of f(x) as x approaches a from the right is L.” As x gets closer and closer to a from the right (x > a), the values of f(x) get closer and closer to L.

3. Copyright © 2011 Pearson Education, Inc. Slide 12.3-3 12.3 One-Sided Limits The left-hand limit, is read “the limit of f(x) as x approaches a from the left is L.” As x gets closer and closer to a from the right (x < a), the values of f(x) get closer and closer to L.

4. Copyright © 2011 Pearson Education, Inc. Slide 12.3-4 12.3 Finding One-Sided Limits of a Piecewise-Defined Function Example Find and where

5. Copyright © 2011 Pearson Education, Inc. Slide 12.3-5 12.3 Finding One-Sided Limits of a Piecewise-Defined Function Solution Since x > 2 in use the formula. In the limit, where x < 2, use f(x) = x + 6.

6. Copyright © 2011 Pearson Education, Inc. Slide 12.3-6 12.3 Infinity as a Limit A function may increase without bound as x gets closer and closer to a from the right

7. Copyright © 2011 Pearson Education, Inc. Slide 12.3-7 12.3 Infinity as a Limit The right-hand limit does not exist but the behavior is described by writing If the values of f(x) decrease without bound, write The notation is similar for left-handed limits.

8. Copyright © 2011 Pearson Education, Inc. Slide 12.3-8 12.3 Infinity as a Limit Summary of infinite limits

9. Copyright © 2011 Pearson Education, Inc. Slide 12.3-9 12.3 Finding One-Sided Limits Example Find and where Solution From the graph

10. Copyright © 2011 Pearson Education, Inc. Slide 12.3-10 12.3 Finding One-Sided Limits Solution and the table and

11. Copyright © 2011 Pearson Education, Inc. Slide 12.3-11 12.3 Limits as x Approaches +  A function may approach an asymptotic value as x moves in the positive or negative direction.

12. Copyright © 2011 Pearson Education, Inc. Slide 12.3-12 12.3 Limits as x Approaches +  The notation, is read “the limit of f(x) as x approaches infinity is L.” The values of f(x) get closer and closer to L as x gets larger and larger.

13. Copyright © 2011 Pearson Education, Inc. Slide 12.3-13 12.3 Limits as x Approaches +  The notation, is read “the limit of f(x) as x approaches negative infinity is L.” The values of f(x) get closer and closer to L as x assumes negative values of larger and larger magnitude.

14. Copyright © 2011 Pearson Education, Inc. Slide 12.3-14 12.3 Finding Limits at Infinity Example Find and where Solution As the values of e -.25x get arbitrarily close to 0 so

15. Copyright © 2011 Pearson Education, Inc. Slide 12.3-15 12.3 Finding Limits at Infinity Solution As the values of e -.25x get arbitrarily large so

16. Copyright © 2011 Pearson Education, Inc. Slide 12.3-16 12.3 Finding Limits at Infinity Solution (Graphing calculator)

17. Copyright © 2011 Pearson Education, Inc. Slide 12.3-17 12.3 Limits as x Approaches +  Limits at infinity of For any positive real number n, and

18. Copyright © 2011 Pearson Education, Inc. Slide 12.3-18 12.3 Finding a Limit at Infinity Example Find Solution Divide numerator and denominator by the highest power of x involved, x 2.

19. Copyright © 2011 Pearson Education, Inc. Slide 12.3-19 12.3 Finding a Limit at Infinity Solution

20. Copyright © 2011 Pearson Education, Inc. Slide 12.3-20 12.3 Finding a Limit at Infinity Solution