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C1 Intuitive Concepts of Limits, Continuity and Differentiability.

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Presentation on theme: "C1 Intuitive Concepts of Limits, Continuity and Differentiability."— Presentation transcript:

1

2 C1 Intuitive Concepts of Limits, Continuity and Differentiability

3 Section 3 Limit of a Function An introduction to limits from a numerical point of view

4 Intuitive Definition. Let y = f(x) be a function. Suppose that a and L are numbers such that whenever x is close to a but not equal to a, f(x) is close to L; as x gets closer and closer to a but not equal to a, f(x) gets closer and closer to L; and suppose that f(x) can be made as close as we want to L by making x close to a but not equal to a. Then we say that the limit of f(x) as x approaches a is L and we write

5 NumericalNumerical and Graphical Exploration 1Graphical

6 Numerical Numerical and Graphical Exploration 2Graphical Don’t trust your calculators!

7 Numerical Numerical and Graphical Exploration 3Graphical Need for considering both right-hand and left-hand limits. Read p.11

8 Find Limits by graphs

9 Example 3.1 Discuss and sketch the graph ofgraph

10 Section 4 Infinity Type 1 A function f(x) tends to positive infinity (respectively negative infinity) as x tends to a if f(x) increases ( respectively decreases) without bounds as x tends to a in any manner. In symbol, we write lim x→a f(x)=+∞ (respectively lim x→a f(x)= -∞)

11 Example 1

12 Example 2

13 Example 3

14 Type 2 A function f(x) approaches to L as x approaches to +  (or -  ) In symbol,

15 Example 4

16 Type 3 f(x) approaches to a limit L as the absolute value of x increases without bounds. In symbol,

17 Example 5

18 Example 6

19 By intuitive concept,

20 On the other hand,

21 Section 5 Evaluation of Limits Limit Theorems

22 Examples

23 Section 6 Continuity of a Function Definition 6.1 A function f is continuous at x = a if and only if

24 Discussion on Ex.1.6 Q.1

25 1.Exercise on ContinuityExercise on Continuity 2.Exercise on the Continuity of piecewise defined functions.Exercise on the Continuity of piecewise defined functions (with animation)

26 Illustrative Examples 6.1 State the points of discontinuity of the following functions: (a) f(x) = tanx (b) g(x) = x – [x] (c) h(x) = (x 2 – 4)/(x+2) x=(2n-1)π/2, nεZ xεZ X= -2

27 Example 6.2 Is the function continuous at x=0? If not, what should be f(0) so that f(x) is continuous there. Solution: ∵ lim x→0 f(x)=lim x→0 sinx/x = 1 and f(0) = 0≠1 ∴ f(x) is not continuous at x = 0. In order that f(x) is continuous at x = 0, f(0) = 1. Ex.1.6, Q.2,3


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