Calculus Problems With Solutions

Calculus is the study of the functions at a particular value or at a nearest time.

To study calculus following topics will be looked into details: ·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 give a value to x a little more than x=1, then we 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.
Lets 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.