# LIMITS OF FUNCTIONS AND DIFFERENTIATION FROM THE FIRST PRINCIPLE

The followings are the  properties of limits:

(i)        lim k = k i e

x2 a

The limit of a constant is the constant itself

(ii)      lim [f(x) + f 2 (x) + f(x) + … fn(x)]

= lim f1(x) + lim f(x) + lim f(x) +limfn(x)

x a        x a       x ax a

i.e

The limit of the sum of a finite number of functions is equal to the sum of their respective limits

lim [f1(x) – f2(x)] = limf1(x) – limf2(x)

x a    x a    x a  x a

EVALUATION

Evaluate lim -> 4   x3 +4x 6

Evaluastelim  x -> -2  x+6/ 2x +4

Differentiation From first Principle

The technique adopted in unit 11.3 in finding the derivative of a function from the consideration of the limiting value is called differentiation from first principle.

Example

Find the derivative of f(x) = x2 from first principle.

Solution

f (x) = x2

f(x + x) = ( x + x)2

= x+ 2x x + ( x)2

f(x + x) – f (x) = (x + x)2 – x2

= x2+ 2x x + ( x)2 – x2

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