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In this section, we will investigate the idea of the limit of a function and what it means to say a function is or is not continuous.

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Presentation on theme: "In this section, we will investigate the idea of the limit of a function and what it means to say a function is or is not continuous."— Presentation transcript:

1 In this section, we will investigate the idea of the limit of a function and what it means to say a function is or is not continuous.

2 The limit of f as x approaches a is L means that f(x) can be made arbitrarily close to L by choosing x values sufficiently close to a. That is, the closer x gets to a (from either side), the closer y will get to L. Note: f(a) does not have to exist, and if it does, it is not necessarily true that f(a) = L.

3 Consider the graph below of y = f(x).

4

5 Consider the function.

6 xf(x) 1.94.9 1.994.99 1.9994.999 xf(x) 2.15.1 2.015.01 2.0015.001

7 Consider the function. xf(x) 1.94.9 1.994.99 1.9994.999 xf(x) 2.15.1 2.015.01 2.0015.001

8 Consider the function.

9 xf(x) 3.9-0.9 3.99-0.99 3.999-0.999 xf(x) 4.11.1 4.011.01 4.0011.001

10 Consider the function. xf(x) 3.9-0.9 3.99-0.99 3.999-0.999 xf(x) 4.11.1 4.011.01 4.0011.001

11 The right hand limit of f as x approaches a is L means that f(x) can be made arbitrarily close to L by choosing x values sufficiently close to, but larger than, x = a. That is, the closer x gets to a (from from the right), the closer y will get to L. Note: f(a) does not have to exist, and if it does, it is not necessarily true that f(a) = L.

12 The left hand limit of f as x approaches a is L means that f(x) can be made arbitrarily close to L by choosing x values sufficiently close to, but smaller than, x = a. That is, the closer x gets to a (from from the left), the closer y will get to L. Note: f(a) does not have to exist, and if it does, it is not necessarily true that f(a) = L.

13 Consider the graph below of y = f(x).

14

15 Consider the function. xf(x) 3.9-0.9 3.99-0.99 3.999-0.999 xf(x) 4.11.1 4.011.01 4.0011.001

16 Consider the function. xf(x) 3.9-0.9 3.99-0.99 3.999-0.999 xf(x) 4.11.1 4.011.01 4.0011.001

17 if and only if and So both left and right hand limits must agree for the overall limit to have a value.

18 Use the graph of y = f(x) below to find:

19 Algebraically find

20

21 Use the definition of derivative to find for the function

22 A function f is continuous at x = a if. So both the left and right hand limits must exist, the function itself must exist, and all three of these must be equal to each other.

23 Where is f not continuous? Why?

24 Consider the function Is f continuous at x = 4? Support your answer.

25 Find all values of “a” so that is continuous for all values of x.

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