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Limits and Continuity Definition Evaluation of Limits Continuity Limits Involving Infinity.

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Presentation on theme: "Limits and Continuity Definition Evaluation of Limits Continuity Limits Involving Infinity."— Presentation transcript:

1 Limits and Continuity Definition Evaluation of Limits Continuity Limits Involving Infinity

2 Limit a L

3 Limits, Graphs, and Calculators Graph 1 Graph 2

4 c) Find Note: f (-2) = 1 is not involved

5 Answer : 16 Answer : no limit Answer : 1/2 3) Use your calculator to evaluate the limits

6 The right-hand limit of f (x), as x approaches a, equals L written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the right of a. a L One-Sided Limit One-Sided Limits

7 The left-hand limit of f (x), as x approaches a, equals M written: if we can make the value f (x) arbitrarily close to L by taking x to be sufficiently close to the left of a. a M

8 1. Given Find Examples Examples of One-Sided Limit

9 Find the limits: More Examples

10 For the function Bu t This theorem is used to show a limit does not exist. A Theorem

11 Limit Theorems

12 Examples Using Limit Rule Ex.

13 More Examples

14 Indeterminate forms occur when substitution in the limit results in 0/0. In such cases either factor or rationalize the expressions. Ex. Notice form Factor and cancel common factors Indeterminate Forms

15 More Examples

16 The Squeezing Theorem See Graph

17 Limits at Infinity For all n > 0, provided that is defined. Ex. Divide by

18 More Examples

19

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21 Infinite Limits For all n > 0, More Graphs

22 Examples Find the limits

23 Limit and Trig Functions From the graph of trigs functions we conclude that they are continuous everywhere

24 Tangent and Secant Tangent and secant are continuous everywhere in their domain, which is the set of all real numbers

25 Examples

26 Limit and Exponential Functions The above graph confirm that exponential functions are continuous everywhere.

27 Asymptotes

28 Examples Find the asymptotes of the graphs of the functions

29

30

31 Continuity A function f is continuous at the point x = a if the following are true: a f(a)f(a)

32 a f(a)f(a)

33 Definition of continuity The function f is continuous at the number c if Defined at c

34 Types of Discontinuity 1- Removable discontinuity f(a) is defined to be a number other than the value of limit. f(a) is not defined or Exist

35 2- Jump discontinuity

36 3- Infinite discontinuity

37 Discuss the continuity of f(x) at x = 2, where Removable discontinuity Example Solution

38 Discuss the continuity of f(x) at x = 0, where Jump discontinuity Example Solution

39 X=-3 Removable discontinuity X=0 Infinite discontinuity X=5 Jump discontinuity X=2 The function is continuous Example Solution

40 What value should b assigned to make the function continuous at x = 1 Example Solution

41 At which value(s) of x is the given function discontinuous? Continuous everywhere Continuous everywhere except at Example s

42 and Thus h is not cont. at x=1. h is continuous everywhere else and Thus F is not cont. at F is continuous everywhere else

43 Continuous Functions A polynomial function y = P(x) is continuous at every point x. A rational function is continuous at every point x in its domain. If f and g are continuous at x = a, then

44 Intermediate Value Theorem If f is a continuous function on a closed interval [a, b] and L is any number between f (a) and f (b), then there is at least one number c in [a, b] such that f(c) = L. ab f (a) f (b) L c f (c) =

45 Example f (x) is continuous (polynomial) and since f (1) 0, by the Intermediate Value Theorem there exists a c on [1, 2] such that f (c) = 0.


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