Limit:

               A function ‘‘ is said to have limit ‘L‘ if ‘‘ is defined for larger value of ‘‘ as approaches to infinity, f(x) approaches to ‘ L‘.

i.e

For example if f(x) = 1/x

then

  f(1) = 1/1 = 1

 f(10) = 1/10 = 0.1

f(100) = 1/100 = 0.01

f(1000) = 1/1000 = 0.001

as x approaches infinity f(x) approaches 0.

Collectively,

A function f(x) is said to have a limit ‘L‘ if f(x) defined for all values of ‘‘ close to ‘a‘  (need not x = a) and as approaches a

e.g

f(x) = x^2 – 4 / x – 2  , x =/= 2

f(x) = (x+2) (x-2) / (x-2)

f(x) = (x+2) , x =/= 2

as x approaches to 2 , f(x) approaches to 4

Rules for Finding Limits

Theorem 1 (Properties of Limits)

Theorem 2

e.g If p(x) = 3x^2 + 2x + 1

Then  = 3(2)^2 + 2(2) + 1

=17

Theorem 3

If P(x) and Q(x) are polynomials and Q(c) =/= 0, then

e.g

 

Theorem 4 (Sandwich Theorem)

Suppose that g(x) <= f(x) <= h(x) for all x in open interval containing ‘c‘ except possibly at x=c itself.

Suppose also that

e.g Cos x < Sin x / x < 1

so by sandwich theorem 

1-

Theorem (One sided vs Two sided)

A function f(x) has a limit as ‘x‘ approaches to ‘‘ if and only if it has left hand limit and right hand limit at x=c and both are equal

Continuity

Continuity of a function at a Point

(Continuity Test)

A function f(x) is said to be continuous at x=c , c element Df if and only if it meets the following three conditions

The example 2 given above is a perfect example of a continuous function.

A function ‘‘ is continuous in an interval [a,b] if it is continuous for all x element [a,b]

Theorem 1 (Continuity of Algebraic combinations)

If functions ‘‘ and ‘g‘ are continuous at x=c , the following functions are also continuous at x=c 

(1) f+g & f-g

(2) fg

(3) f/g, provided g(c) =/= 0

(4) k where ‘ k’ is any constant

(5) [ f(x) ]^m/n , provided [ f(x) ]^m/n  is defined on an interval containing ‘c’ and m,n are integers

Theorem 2

Every polynomial is continuous at every point of real line. Every rational function is continuous at every point where it’s denominator is not zero.

f(x) = x+1 / x-1

continuous at every point except 1.

Theorem 3

If ‘‘ is continuous at ‘‘ and ‘g‘ is continuous at f(c) then gof  is continuous at x=c.

Types of disContinuity

 Non-Removable discontinuity

 If at any point a function does not have functional value as well as limiting value then the type of discontinuity is called non-removable discontinuity

e.g

f(x)=1/x

f(0) does not exist

Removable disContinuity

If at any point , a function does not have any functional value but it has limiting value at that point then that kind of discontinuity is called removable discontinuity

e.g

f(x) = x^2 – 4 / x-2

f(2) does not exist

Continuous Extension

If f(c) does not exist but    then we can define a new function f(x) as

Then the function f(x) is continuous at x=c . It is called continuous extension of f(x) to x=c.

Example 1

 

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