Limits, Continuity and Differentiability

Absolute Values

Properties of absolute values : If |x| = 0, then x = 0 and conversely |0| = 0. (2) If x 0, then |x| > 0. (3) |x| = |-x| and |a – b| = |b – a|. (4) If |x| =|a|, then x = ; particularly, if |x| = a, then x = .

Some Special Symbols i.e. max{a,b} denotes the larger number between a and b, while min{a,b} denotes the smaller number between a and b.

The graphs of y = max {f(x), g(x)} and y = min {f(x), g(x)} Some Special Symbols The graphs of y = max {f(x), g(x)} and y = min {f(x), g(x)}

Consider the function Limit of a Function x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f(x)

Limit of a Function x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f(x) 3.9 3.99 3.999 3.9999 un-defined 4.0001 4.001 4.01 4.1

Consider the function Limit of a Function x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 g(x)

Limit of a Function Consider the function x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1 f(x) 3.9 3.99 3.999 3.9999 4 4.0001 4.001 4.01 4.1 y = x+2

Limit of a Function Consider the function x -4 -3 -2 -1 1 2 3 4 g(x)

Limit of a Function Consider the function x -4 -3 -2 -1 1 2 3 4 g(x) 1 2 3 4 g(x) un-defined

Limit of a Function does not exist.

Limit of a Function Let f(x) be a function of real variable x, the symbol denotes that f(x) tends to L as x approaches (or tends) to a.

Limit of a Function

Limit of a Function

Limit of a Function If a function f(x) possess limit at x = a, i.e. the limit exists, then this limit is independent of the fashion that x tends to a. Whether x tends to a from the right hand side or from left hand side, the two limits obtained must be the same. Hence, in order that exists, the two limits both exist and equal. i.e. ,where L is a real number.

Limit of a Function Concept of limit is illustrated by the following diagrams :

Limit of a Function Concept of limit is illustrated by the following diagrams :

Concept of limit is illustrated by the following diagrams : Limit of a Function Concept of limit is illustrated by the following diagrams : (c)

y1

y1

Infinity

Infinity

Infinity

Infinity

Infinity

Infinity y x

(b)

Continuity of a Function f(x) is continuous at x = a means that the graph of y = f(x) has no break at x = a.

Continuity of a Function Types of interval : The closed interval [a,b] is the collection of all real numbers x such that . The open interval (a,b) is the collection of all real numbers x such that . The half open-closed interval (a,b] is the collection of all real numbers such that . The half closed-open interval [a,b) is the collection of all real numbers such that .

Continuity of a Function Types of intervals : (5) The interval (-, + ) is the collection of all real numbers. (6) The interval (- , a) is the collection of all the real numbers x such that x < a. (7) The interval (a, + ) is the collection of all the real numbers x such that x > a. (8) The intervals (- , a] and [a, + ) are defined similarly.

Continuity of a Function 5 kinds of continuity :

Continuity of a Function 5 kinds of continuity :

3 kinds of discontinuity : Continuity of a Function A function f(x) is said to be discontinuous at x = a if it is not continuous at x = a. 3 kinds of discontinuity :

Differentiability of a Function

f(x) is not differentiable at x = x0 Differentiability of a Function x y x0 P A B f(x) is not differentiable at x = x0

f(x) is differentiable at x = x0 Differentiability of a Function B x y P A x0 f(x) is differentiable at x = x0

f(x) is not differentiable at x = x0 Differentiability of a Function x y A P B x0 f(x) is not differentiable at x = x0

Differentiability of a Function If a function f(x) is differentiable at x = x0, then it must be continuous there, or we may say that if f(x) is not continuous at x = x0, it must not be differentiable there.