DIFFENTIAL CALCULUS

* Define differential calculus

*Explain the concept of limit of a function.

*State the properties of limits.

DEVELOPMENT.

The branch of mathematics dealing with continuous varying quantities is called calculus,while differential calculus deals with rate of change.

LIMIT OF A FUNCTION.

Limit of a function can be defined as the definite value arrived at as a result of small increament in the value of a given function.

If the limit of functions is from the right , we called right-hand -limit and if it is from the left, it is called left hand limit.The concept of limit  is so fundamental  in differential calculus  that it cannot be glossed over.

Example (1) Consider the behaviour of the simple quadratic function f (x)=x^2 at x=3.1 , 3.01, 3.001, 3.0001,— – – -3+ 10^n. See the table below.

From the above table, we observe that as sequence of f (x) number increases, the corresponding sequence generated from f (x)=x^2 come closer  to 9. Thus the limiting  value of f (x) as x approaches 3 from left to right is 9.

PROPERTY  OF LIMITS.

1. The limit of constant is constant itself.

2. The limit of a sum of finale number of functions is the sum  of their respective  limits.

3. The limit of a difference of two functions  is the difference  of their limits.

4. The limit  of the product of a finite number of functions  is equal to the product of their respective  limits.

5. The limit of quotient of two functions is the quotient of their limits,provided the limit of the denominator is not equal to zero.

6.The limit of the product of a constant and a function is equal to the product of the constant and the limit of the function.

EXAMPLE 2 . Evaluate lim(6^3 – 3x^2 +5x +7 )

                                     x~0.

SOLUTION :lim6x^3 -lim3x^2 +lim 5x  +lim7 .

                      x~0

=0 -0 +0 + 7

=7.

EXAMPLE 3 Evaluate: limx~0  (x -2)(3x -4)(2x +5).

SOLUTION :

lim x~0 (x -2)*lim (3x – 4)*lim (2xx + 5).

=(-2) (-4) (5)

=40.

EVALUATION :

1. Explain the concept of limit of a function.

2 List five properties of limits of functions.

3. Evaluate lim x~3 (2x^2 + 3x – 4).