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**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**

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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**Numerical and Graphical Exploration 1**

After working through these materials, the student should be able to obtain numerical evidence for the calculation of limits; to determine what appears to be the limit from the numerical evidence; and to become aware of some of the problems in using numerical evidence for the calculation of limits.

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**Numerical and Graphical Exploration 2**

Don’t trust your calculators! Numerical and Graphical Exploration 2 After working through these materials, the student should be able to obtain numerical evidence for the calculation of limits; to determine what appears to be the limit from the numerical evidence; and to become aware of some of the problems in using numerical evidence for the calculation of limits.

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**Numerical and Graphical Exploration 3**

Read p.11 Numerical and Graphical Exploration 3 After working through these materials, the student should be able to obtain numerical evidence for the calculation of limits; to determine what appears to be the limit from the numerical evidence; and to become aware of some of the problems in using numerical evidence for the calculation of limits. Need for considering both right-hand and left-hand limits.

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Find Limits by graphs

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Example 3.1 Discuss and sketch the graph of

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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 limx→af(x)=+∞ (respectively limx→af(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 Continuity**

2.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) = (x2 – 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: ∵limx→0f(x)=limx→0sinx/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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Limits Involving Infinity Section 1.4. Infinite Limits A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite.

Limits Involving Infinity Section 1.4. Infinite Limits A limit in which f(x) increases or decreases without bound as x approaches c is called an infinite.

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