Sections in the Chapter
21.1 Introduction
21.2 Continuity at a point
21.3 Continuity functions in an interval
21.4 Continuous functions
21.5 Cauchy’s definition of continuity
21.6 Heine’s definition of continuity
21.7 Discontinuous functions
21.8 Properties of continuous functions
21.9 Differentiability at a point
21.10 Relation between continuity and differentiability
21.11 Differentiability in a set
21.12 Some results on differentiability
Study Plan
Day 1
21.1 Introduction
21.2 Continuity at a point
21.3 Continuity functions in an interval
21.4 Continuous functions
21.5 Cauchy’s definition of continuity
Day 2
21.6 Heine’s definition of continuity
21.7 Discontinuous functions
21.8 Properties of continuous functions
Day 3
21.9 Differentiability at a point
21.10 Relation between continuity and differentiability
21.11 Differentiability in a set
21.12 Some results on differentiability
Day 4
Objective Type Exercises: 1 to 20
Day 5
O.T.E.: 21 to 40
Day 6
O.T.E.: 41 to 60
Day 7
O.T.E.: 61 to 80
Day 8
O.T.E.: 81 to 100
Day 9
O.T.E.: 101 to 127
Day 10
Fill in the blank Exercises 1 to 21
Day 11
True/False Type Exercises 1 to 11
Day 12
Practice Exercises 1 to 7
Day 13
Practice Exercises 8 to 14
21.1 Introduction
21.2 Continuity at a point
21.3 Continuity functions in an interval
21.4 Continuous functions
21.5 Cauchy’s definition of continuity
21.6 Heine’s definition of continuity
21.7 Discontinuous functions
21.8 Properties of continuous functions
21.9 Differentiability at a point
21.10 Relation between continuity and differentiability
21.11 Differentiability in a set
21.12 Some results on differentiability
Showing posts with label Limits and continuity. Show all posts
Showing posts with label Limits and continuity. Show all posts
Friday, November 7, 2008
Thursday, June 5, 2008
Chapter 20 Limits - 1
Material covered in class XIth book.
Limit of a function
Limit x→a f(x) = l, means | f(x)-l| can be made as small as we like by making | x-a| sufficiently small without making x = a.
Since we undertake to make | f(x)-l| as small as required, we will be told how small | f(x)-l| is to be made. We will be given a criterion ε>0 and we must make | f(x)-l| < ε for values of x near a. We must do this by making | x-a| small enough. Hence we must find δ>0 such that when | x-a| < δ (and x≠a), | f(x)-l| will be less than ε.
| f(x)-l| < ε means f(x) belong to the ε neighbourhood of l and | x-a| < δ (and x≠a) means x belongs to the deleted δ neighbourhood of a.
Note:
The open interval (a- δ, a+ δ) whose length is 2 δ and whose midpoint is a, is called the δ-interval of a. Open interval means a- δ, and a+ δ are not part of the interval.
{x/x Є(a- δ,a+ δ), x≠0} is called the deleted δ-neighbourhood of a.
Definition of limit
Lim x→a f9x) = 1 if, given any ε>0, we can find δ>0 such that | f(x)-l| < ε whenver 0<| x-a| < δ.
Algebra of limits
1. lim x→a [f(x) ±g(x)] = lim x→a f(x) ±lim x→ag(x)
2. lim x→a [k.f(x)] = k lim x→af(x)
3. lim x→a[f(x).g(x)] = [lim x→a f(x)][ lim x→a g(x)]
4. lim x→a [f(x)/g(x)] = [lim x→a f(x)]/[ lim x→ag(x)] provided lim x→a g(x) ≠
0.
(Note: most of the problems given in exercises in the chapter are on applying this condition. Check whether the denominator becomes zero at ‘a’ and then check if the numerator also becomes zero at the ‘a’. If both become zero, it means there is a common factor and one has to remove the common factor to find the limit)
5. If f(x)
Standard limits
1. lim x→a x = a
2. lim x→a kx = ka, k Є R
3. lim x→a xk = ak , k Є R
4. lim x→a k = k
5. lim x→a (xk –ak )/(x-a) = nan-1, nk Є N
6. lim x→a sin x = sin α
7. lim x→a cos x = cos α
8. lim x→a (sin x)/x = 1, where x is measured in radians
Limit of a function
Limit x→a f(x) = l, means | f(x)-l| can be made as small as we like by making | x-a| sufficiently small without making x = a.
Since we undertake to make | f(x)-l| as small as required, we will be told how small | f(x)-l| is to be made. We will be given a criterion ε>0 and we must make | f(x)-l| < ε for values of x near a. We must do this by making | x-a| small enough. Hence we must find δ>0 such that when | x-a| < δ (and x≠a), | f(x)-l| will be less than ε.
| f(x)-l| < ε means f(x) belong to the ε neighbourhood of l and | x-a| < δ (and x≠a) means x belongs to the deleted δ neighbourhood of a.
Note:
The open interval (a- δ, a+ δ) whose length is 2 δ and whose midpoint is a, is called the δ-interval of a. Open interval means a- δ, and a+ δ are not part of the interval.
{x/x Є(a- δ,a+ δ), x≠0} is called the deleted δ-neighbourhood of a.
Definition of limit
Lim x→a f9x) = 1 if, given any ε>0, we can find δ>0 such that | f(x)-l| < ε whenver 0<| x-a| < δ.
Algebra of limits
1. lim x→a [f(x) ±g(x)] = lim x→a f(x) ±lim x→ag(x)
2. lim x→a [k.f(x)] = k lim x→af(x)
3. lim x→a[f(x).g(x)] = [lim x→a f(x)][ lim x→a g(x)]
4. lim x→a [f(x)/g(x)] = [lim x→a f(x)]/[ lim x→ag(x)] provided lim x→a g(x) ≠
0.
(Note: most of the problems given in exercises in the chapter are on applying this condition. Check whether the denominator becomes zero at ‘a’ and then check if the numerator also becomes zero at the ‘a’. If both become zero, it means there is a common factor and one has to remove the common factor to find the limit)
5. If f(x)
Standard limits
1. lim x→a x = a
2. lim x→a kx = ka, k Є R
3. lim x→a xk = ak , k Є R
4. lim x→a k = k
5. lim x→a (xk –ak )/(x-a) = nan-1, nk Є N
6. lim x→a sin x = sin α
7. lim x→a cos x = cos α
8. lim x→a (sin x)/x = 1, where x is measured in radians
Monday, April 28, 2008
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