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C1 Intuitive Concepts of Limits, Continuity and Differentiability

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

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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

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NumericalNumerical and Graphical Exploration 1Graphical

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

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

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Find Limits by graphs

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Example 3.1 Discuss and sketch the graph ofgraph

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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)= -∞)

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Example 1

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Example 2

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Example 3

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

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Example 4

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

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Example 5

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Example 6

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By intuitive concept,

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On the other hand,

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Section 5 Evaluation of Limits Limit Theorems

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Examples

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

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Discussion on Ex.1.6 Q.1

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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)

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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

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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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