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# Introduction of Calculus

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 Sub Topics Calculus is the study of the Functions at a particular value or at a nearest time. To learn Calculus following topics should be studied in detail: ·Limits ·Limits & Continuity ·Differentiation ·Integration To start up, we first talk about Limits. Let us consider a function: f(x) =( x^2 -1 ) / (x - 1) Here, if we place the value of x=1, then f(1)= (1^2 -1) / (1-1) = 0 / 0 , which is not meaningful value. We know that ( x^2-1 ) = (x - 1) (x+1) So , f(x) =(x^2-1) / (x - 1) = (x+1) (x-1) /(x-1) Cancelling (x-1) from numerator and denominator, we get f(x) = (x+1), only if x ≠1. Further, let us imagine that we are giving a value to x a little more than x=1, then we can observe that the value of f(x) will be a little more than 2. Slowly if we go on sliding the value of x nearer to 1, but not exactly 1, then the value of function f(x) will go on sliding towards 2. Let’s see how this change occurs: If we take x=1.1, then the value of f(x) =2.1, If we take x= 1.01, then the value of f(x)= 2.01, If we take x= 1.001, then the value of f(x)= 2.001, Proceeding in the same way, If we take x=1.0000001, then the value of f(x)= 2.0000001 We conclude that as the value of x approaches 1, the value of f(x) approaches to 2. We write it as: x→1, then f(x) →2 Now let us try to move towards another direction and observe the change. Let us take the value of x a little lesser than 1, we find that the value of f(x) is a little lesser than 2. So, If we take x=0.9, then the value of f(x) =1.9, If we take x= 0.09, then the value of f(x)= 1.09, If we take x= 0.009, then the value of f(x)= 1.009, Proceeding in the same way, If we take x=0.0000009, then the value of f(x)= 1.0000009 We conclude that as the value of x approaches 1, the value of f(x) approaches to 2. We write it as: x→1, then f(x) →2 lim f(x)=m, means if x →a, f(x) →m. x→a While finding the limits of a given function, certain rules are to be remembered. They are as follows: · We simply put the value x=a in a given function and check if f(x) is a definite value, then simply Lim f(x) = f(a) x→a · In case, we find f(x) as a rational number, then we simply factorize the numerator and the denominator, then cancel the common factors and finally place the value of x=a. · In case, we find the given function contains a surd, then we simplify the function by multiplying the numerator and the denominator of the given function with the conjugate of the given surd. Then we simplify it and finally put x=a in it to get the solution. · In case, a function contains a series, which can be expanded, then expand it, simplify it, cancel the common factors of the numerator and denominator and finally put the value x=a to get the solution.