LIMITS OF FUNCTIONS AND DIFFERENTIATION FROM THE FIRST PRINCIPLE

Back to: Further Mathematics SS2

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_3 (x) + … f_n(x)]

= lim f₁(x) + lim f₂ (x) + lim f₃ (x) +limf_n(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

= x² + 2x x + ( x)²

f(x + x) – f (x) = (x + x)² – x²

= x²+ 2x x + ( x)² – x²

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