The Limits and Derivatives are an integral part of calculus. We can say that limit and derivative are part of integration and differentiation respectively. In the field of calculus, the Limit of any function is defined as a function f(x) which has value ‘P’, and ‘x’ approaches ‘a’. We can also express it by following term:
limx->a f(x) = P,
This means the limit function is responsible to make the function f(x) as close to ‘p’ as possible, by making choice on ‘x’ in a very small interval around ‘a’. (Following the condition x -> a)
By the above expression we mean that value of function f(x) reaches ‘P’ if and only if value of ‘x’ is close to the value of ‘a’.
The limit of any function is calculated by some laws which are called by the name called limit laws and some important laws are as given bellow:
If limit functions limx->a c(x) and limx->a m(x) are exist then we have,
1. limx->a c(x) +- m(x) = limx->a c(x) +- limx->a m(x),
2. limx->a c(x) m(x) = limx->a c(x) . limx->a m(x),
3. limx->a c(x) / m(x) = limx->a c(x) / limx->a m(x), here the limit limx->a m(x) is not equals to zero.
We know that algebraic functions consist of polynomials and we can use the elementary algebraic operations by taking roots.
Assume g(x) is an algebraic function and f (b) is defined, then limx->b g(x) = g (b). Here we mean that calculation of limit is also possible by substituting ‘b’ in place of ‘x’.
Derivate is defined as the differentiation of any function. We can also understand the differentiation by the following expression.
Differentiation of function f(x) is:
f’(x) or d/dx f(x).
The derivative is the study of how the derivative function changes as its input changes. In other words the process to determine the derivative is differentiation and the reverse of this process is called anti-differentiation.
Here dy / dx is said to be positive if there is an increment in the variable (y) with respect to the increment in the variable (x), and the derivative values are said to be negative if there is a decrement in the variable (y) with respect to the value of variable (x).
Suppose (f) is a function having a derivative (p) at each and every point in the domain of ‘f’, because each and every point of (a) has a derivative.
This is all about limits and derivatives.