Mathematics Class XI

Unit-I: Sets and Functions
Chapter 1: Sets
Unit-II: Algebra
Chapter 5: Binomial Theorem
Chapter 6: Sequence and Series
Unit-III: Coordinate Geometry
Chapter 1: Straight Lines
Chapter 2: Conic Sections
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability
Chapter 1: Statistics
Chapter 2: Probability

Limits and their Applications

Lesson Progress
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Convergence of a sequence:

A number is called the limit of sequence if given , there exists such that for . If is the limit of a sequence , then we write

Definition:

A sequence is said to be convergent if exists, otherwise it is called divergent.

If , then we say that the sequence converges to .

Note:

A sequence converges to  iff every sub-sequence of converges to .

Theorem:

for every real sequence with . For any and .

Limits of Trigonometry Functions-:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

(xiii)

(xiv)

(xv)

(xvi)

(xvii)

(xviii)

if

(xix)

Limits of Exponential and Logarithmic Functions:

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi) If , then

(xii) If then,

Monotonic Sequence:

Definition:

Let be a real sequence:

(i) The sequence is said to be monotonic increasing or non-decreasing if .

(ii) If , then the sequence is strictly increasing.

(iii) If , then the sequence is monotonic decreasing.

(iv) If , then the sequence is strictly decreasing.

Bounded Sequence:

(i) A real sequence is said to be bounded above if there exists a real number such that . The number is called an upper bound of the sequence.

(ii) is said to be bounded below if there exists a real number such that , then is called lower bound of the sequence.

(iii) If there exists a real number such that  , then the sequence is said to be bounded.

A sequence is said to be Bounded iff it is bounded above and below.

Least Upper Bound (LUB) (Supremum)

A real sequence is said to be bounded above i.e. upper bounds of exist and denoted as , where The least upper bound (LUB) or Supremum of a sequence is said to be the least number exist in the upper bound set of . i.e. .

Greatest Lower Bound (GLB) or (Infimum)

A real sequence is said to be bounded below i.e. lower bounds of exist and denoted by , where .

The greatest lower bound (GLB) or infimum of a sequence is said to be the greatest number exist in the lower bound set of i.e. .

Theorem:

(i) A monotonic increasing sequence bounded above is convergent, i.e. tends to a limit as

(ii) A monotonic decreasing sequence bounded below is convergent, i.e. tends to a limit as

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