Intuitive Concepts of Limits, Continuity and Differentiability

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

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

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.

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.

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.

Find Limits by graphs

Example 3.1 Discuss and sketch the graph of

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)= -∞)

Example 1

Example 2

Example 3

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

Example 4

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

Example 5

Example 6

By intuitive concept,

On the other hand,

Section 5 Evaluation of Limits Limit Theorems

Examples

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

Discussion on Ex.1.6 Q.1

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

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

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