Limit of a function concept

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Limit of a function concept

Limit of a function concept is divided into two part

(1) descriptively

(2) numerically

Limit of a function concept

 

(1) By finding the area of circumscribing regular polygon:

Consider a circle of unit radius which circumscribes a square (4-sided regular polygon) as shown in figure 1.

The side of the square is √ 2 and area is 2 square unit. It is clear that the area of the inscribed 4-sided polygon is less than the area of the cir cum circle.

Bisecting the Arc between the vertices of the square, we get the inscribed 8-sided polygon, as shown in figure  2. Its area is 2 √ 2 square unit. Which is closer to the area of cir cum-circle.

A further similar bisection of the arcs gives an inscribed 16-sided polygon, as shown in figure 3 with area 3.061 square unit. Which is more close to the area of cir cum-circle.

It follows that as’ n ‘the number of the sides of the inscribed polygon increases, the area of the polygon increases and becoming nearer to 3.142.

Which is the area of the circle of unit radius.

That is

π r² =π (1)²  = π = 3.142.

We express this situation by stating that the limiting value of the area of the inscribed polygon is the area of the circle as n approaches to infinity

that is

Area of inscribe polygon → Area of circle

As n → ∞

Thus, area of the circle of the limit   = π = 3.142 (approximately)

(2) Numerically approaches on circle (Limit of a function concept Numerically)

consider the function f(x) = x ³

The domain of f(x) is the set of all real numbers.

Let us fined the limit of f(x) = x³ as x approaches 2.

The table of f(x) for different value of x as x approaches to 2 from left and right is as follows.

x approaches to infinity means

The table shows that, as x gets closer and closer to 2 (sufficiently close to 2) from both sides , f(x) gets closer and closer to 8.

We say that 8 is the limit of f(x) when x approaches to 2 and is written as :

f(x)  →  8,

  as x → 2,

  OR 

  Lim x → 2(x³)  = 8

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