Limits

“nearness”

Consider a polygon inscribed in a circle

The Idea of Limits

n=3 n=4 n=5 n=6 n=7 n=8‘As number of sides of polygon increases, its area

approximates the area of the circle’‘limit of Area of polygon is the Area of the circle’

As n approaches infinity , Lim Area of polygon = Area of the circle

The Idea of Limits

Consider the function:

The Idea of Limits 2)( xxg

2)( xxg

x

y

O

2

x 1.9 1.99 1.999 1.9999 2 2.0001 2.001 2.01 2.1

g(x) 3.9 3.99 3.999 3.9999 3.0001 4.001 4.01 4.14

4)(lim2

xgx

As x approaches to positive 2 at both directions

Fundamental Rules of Limits.1. The Constant Rule

– When we take the limit of a constant, non-changing function, the limit will simply be that constant.

2. The Sum Rule– If two sequences have limits that exist, then the limit of

the sum of sequences is the sum of the limits of the sequences.

3. The Multiplication Rule– If two sequences have limits that exist, then the limit of

the product is the product of the limits.

Fundamental Rules of Limits.

Techniques in calculating Limits

T1: Limits By Direct Substitution

T2: Limits by Factoring

Type 3a: Limits by Rationalization

Techniques in calculating Limits

T3b: Limits by Rationalization

Techniques in calculating Limits

T4a: Limits at Infinity

Techniques in calculating Limits

T4b: Limits at Infinity

Techniques in calculating Limits

T5: Trigonometric Limits

Techniques in calculating Limits

T6: Limits Involving Number e

Techniques in calculating Limits

Try me!!

Try me!!

Try me!!

Try me!!

Try me!!

1)(lim0

xhx

1)(lim0

xhx

)(lim0

xhx does not

exist.

Two Sided limit

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