13 min read

Limits And Derivatives

- Let's dig deep into the concepts of the chapter


Let .When we approach the point from the values which are greater than or smaller than , would have tendency to move closer to the value .


Indeterminate Forms

We know that  
Lets consider a very interesting case ,
This case is absurd and  can be investigated by this odd argument
If then clearly
So if then clearly
If we choose any two numbers instead of and we will get the same answer.
For eg:
Lets take and instead off and .

This brings us to the conclusion that can be equal to any number
The above conclusion is absurd.
Hence,we call as Indeterminate Form

Concept of Limits

Lets understand the concept of limits with the help of a function,

Let's work it out for :

Now is a difficulty! We don't really know the value of since,it is indeterminate, so we need another way of answering this.

So instead of trying to find value of at let's try approaching  closer and closer to as shown.


We are now faced with an interesting situation:

  • When we don't know the value of (it is indeterminate)
  • But we can see that it is approaching the value 2 when is approaching

We summarize the behaviour of  by using special word "limit"

We say that the limit of , as approaches is 

And it is written in symbols as:


   since,    we can also represent it as,


So it is a special way of saying, "ignoring what happens to the function when  , but as we get closer and closer to  the value of the function gets closer and closer to


Algebra of Limits

1.Limit of sum of two functions is the sum of the limits of the function.

2.Limit of difference of two functions is difference of the limits of the function.

3. Limit of product of two functions is product of the limits of the function.

4. Limit of quotient of two functions is quotient of the limits of the function. (if denominator is non zero).

(if )

Limit of A Polynomial Function.

we first put in


Procedure to Find Limits of A Rational Function.

Any Function of the type (where and are polynomials, and ) is known as Rational Function.
To Solve Limits of a rational function.
  1. Put in , if we get then proceed to next step.Else find the answer by direct substitution.
  2. Factorize and and cancel out the non zero common terms.
  3. Repeat Step 1 and Step 2 untill we get a real number as an answer.

Limits of Trigonometric Functions Using Direct Substitution.

We first put in


Sandwich Theorem

If , and are three functions such that for all in some interval containing the point  
such that  


Theorems on Differentiability

If and are both derivable at , , and will also be derivable at .

If both and are non derivable , then nothing can be said about the sum /difference /product function.
if is derivable at and and is continuous at

Trigonometric Functions



Derivative of sum/difference of functions

If and are both derivable at , ,
and will also be derivable at

Leibinitz Rule

and are functions of


Quotient Rule

Given functions of and