A function ‘f ‘ is said to have limit ‘L‘ if ‘f ‘ is defined for larger value of ‘x ‘ as approaches to infinity, f(x) approaches to ‘ L‘.
For example if f(x) = 1/x
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.
A function f(x) is said to have a limit ‘L‘ if f(x) defined for all values of ‘x ‘ close to ‘a‘ (need not x = a) and as approaches a
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)
e.g If p(x) = 3x^2 + 2x + 1
Then = 3(2)^2 + 2(2) + 1
If P(x) and Q(x) are polynomials and Q(c) =/= 0, then
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
Theorem (One sided vs Two sided)
A function f(x) has a limit as ‘x‘ approaches to ‘c ‘ if and only if it has left hand limit and right hand limit at x=c and both are equal
Continuity of a function at a Point
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 ‘f ‘ 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 ‘f ‘ and ‘g‘ are continuous at x=c , the following functions are also continuous at x=c
(1) f+g & f-g
(3) f/g, provided g(c) =/= 0
(4) kf 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
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.
If ‘f ‘ is continuous at ‘c ‘ and ‘g‘ is continuous at f(c) then gof is continuous at x=c.
Types of 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
f(0) does not exist
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
f(x) = x^2 – 4 / x-2
f(2) does not exist
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.