# NCERT Solutions for Class 11 Maths Chapter 13 Limits and Derivatives

NCERT solutions for class 11 maths chapter 13 Limits and Derivatives: This chapter introduces an important area of mathematics called calculus. This is a branch of mathematics which deal with the study of change in the value of a function as the points in the domain change.  In this article, you will get NCERT solutions for class 11 maths chapter 13 limits and derivatives. This chapter starts with an intuitive idea of derivative (without actually defining it) then covers the definition of a limit, algebra of limits, the definition of derivative, and algebra derivatives. You will also learn derivatives of certain standard functions. In solutions of NCERT for class 11 maths chapter 13 limits and derivatives, you will get questions related to all the above topics. This chapter is very important for class 11 final examination and also in various competitive exams like JEE Main, JEE Advanced, VITEEE, BITSAT, etc. The knowledge of this chapter is required to study any chapter of calculus that you will be studying in the upcoming class 12 also. There are 43 questions in 2 exercises. The first exercise of this chapter deals with problems on limits and the second exercise deals with problems on derivatives. All these questions are solved in the CBSE NCERT solutions for class 11 maths chapter 13 limits and derivatives. Check all NCERT solutions from class 6 to 12 to understand the concepts in a much easy way. There are two exercises and a miscellaneous exercise of this chapter are explained below.

Exercise:13.1

Exercise:13.2

Miscellaneous Exercise

Topics of NCERT Grade 11 Maths Chapter-13 Limits and Derivatives

13.1 Introduction

13.2 Intuitive Idea of Derivatives

13.3 Limits

13.4 Limits of Trigonometric Functions

13.5 Derivatives

## NCERT solutions for class 11 maths chapter 13 limits and derivatives-Exercise: 13.1

Question:1 Evaluate the following limits

Question:2 Evaluate the following limits

Below you can find the solution:

Question:3 Evaluate the following limits

The limit

Question:4 Evaluate the following limits

The limit

Question:5 Evaluate the following limits

The limit

Question:6 Evaluate the following limits

The limit

Lets put

since we have changed the function, its limit will also change,

so

So our function have became

Now As we know the property

Hence,

Question:7 Evaluate the following limits

The limit

Question:8 Evaluate the following limits

The limit

At x = 2 both numerator and denominator becomes zero, so lets factorise the function

Now we can put the limit directly, so

Question:9 Evaluate the following limits

The limit,

Question:10 Evaluate the following limits

The limit

Here on directly putting limit , both numerator and the deniminator becomes zero so we factorize the function and then put the limit.

Question:11 Evaluate the following limits

The limit:

Since Denominator is not zero on directly putting the limit, we can directly put the limits, so,

Question:12 Evaluate the following limits

Here, since denominator becomes zero on putting the limit directly, so we first simplify the function and then put the limit,

Question:13 Evaluate the following limits

The limit

Here on directly putting the limits, the function becomes  form. so we try to make the function in the form of . so,

As

Question:14 Evaluate the following limits

The limit,

On putting the limit directly, the function takes the zero by zero  form  So,we convert it in the form of .and then put the limit,

Question:15 Evaluate the following limits

The limit

Question:16 Evaluate the following limits

The limit

the function behaves well on directly putting the limit,so we put the limit directly. So.

Question:17 Evaluate the following limits

The limit:

The function takes the zero by zero form when the limit is put directly, so we simplify the function and then put the limit

Question:18 Evaluate the following limits

The function takes the form zero by zero when we put the limit directly in the function,. since function consist of sin function and cos function, we try to make the function in the form of  as we know that it tends to 1 when x tends to 0.

So,

Question:19 Evaluate the following limits

As function doesn't create any abnormality on putting the limit directly,we can put limit directly. So,

Question:20 Evaluate the following limits

The function takes the zero by zero form when we put the limit into the function directly, so we try to eliminate this case by simplifying the function. So

Question:21 Evaluate the following limits

On putting the limit directly the function takes infinity by infinity form, So we simplify the function and then put the limit

Question:22 Evaluate the following limits

The function takes zero by zero form when the limit is put directly, so we simplify the function and then put the limits,

So

Let's put

Since we are changing the variable, limit will also change.

as

So function in new variable becomes,

As we know tha property

Question:23 Find

Given Function

Now,

Limit at x = 0  :

:

Hence limit at x = 0 is 3.

Limit at x = 1

Hence limit at x = 1 is 6.

Question:24 Find

Limit at

Limit at

As we can see that Limit at  is not equal to Limit at ,The limit of this function at x = 1 does not exists.

Question:25 Evaluate

The right-hand Limit or  Limit at

The left-hand limit or Limit at

Since Left-hand limit and right-hand limit are not equal, The limit of this function at x = 0 does not exists.

Question:26 Evaluate

The right-hand Limit or  Limit at

The left-hand limit or Limit at

Since Left-hand limit and right-hand limit are not equal, The limit of this function at x = 0 does not exists.

Question:27 Find

The right-hand Limit or  Limit at

The left-hand limit or Limit at

Since Left-hand limit and right-hand limit are equal, The limit of this function at x = 5 is 0.

Question:28 Suppose

f (x) = f (1) what are possible values of a and b?

Given,

And

Since the limit exists,

left-hand limit = Right-hand limit = f(1).

Left-hand limit  = f(1)

Right-hand limit

From both equations, we get that,

and

Hence the possible value of a and b are 0 and 4 respectively.

Given,

Now,

Hence

Now,

Hence

.

Question:30 If    For what value (s) of a does  exists ?

Limit at x = a exists when the right-hand limit is equal to the left-hand limit. So,

Case 1: when a = 0

The right-hand Limit or  Limit at

The left-hand limit or Limit at

Since Left-hand limit and right-hand limit are not equal, The limit of this function at x = 0 does not exists.

Case 2: When a < 0

The right-hand Limit or  Limit at

The left-hand limit or Limit at

Since LHL = RHL, Limit exists at x = a and is equal to a-1.

Case 3: When a > 0

The right-hand Limit or  Limit at

The left-hand limit or Limit at

Since LHL = RHL, Limit exists at x = a and is equal to a+1

Hence,

The limit exists at all points except at x=0.

Question:31 If the function f(x) satisfies    , evaluate

Given

Now,

Question:32 If

. For what integers m and n does both   exist ?

Given,

Case 1: Limit at x = 0

The right-hand Limit or  Limit at

The left-hand limit or Limit at

Hence Limit will exist at x = 0 when m = n .

Case 2: Limit at x = 1

The right-hand Limit or  Limit at

The left-hand limit or Limit at

Hence Limit at 1 exists at all integers.

## NCERT solutions for class 11 maths chapter 13 limits and derivatives-Exercise: 13.2

Question:1 Find the derivative of

F(x)=

Now, As we know, The derivative of any function at x is

The derivative of f(x) at x = 10:

Question:2 Find the derivative of x at x = 1.

Given

f(x)= x

Now, As we know, The derivative of any function at x is

The derivative of f(x) at x = 1:

Question:3 Find the derivative of 99x at x = l00.

f(x)= 99x

Now, As we know, The derivative of any function at x is

The derivative of f(x) at x = 100:

Given

f(x)=

Now, As we know, The derivative of any function at x is

f(x)=

Now, As we know, The derivative of any function at x is

f(x)=

Now, As we know, The derivative of any function at x is

Given:

Now, As we know, The derivative of any function at x is

As we know, the property,

applying that property we get

Now.

So,

Here

Hence Proved.

Given

As we know, the property,

applying that property we get

Question:7(i) For some constants a and b, find the derivative of

Given

As we know, the property,

and the property

applying that property we get

Question:7(ii) For some constants a and b, find the derivative of

Given

As we know, the property,

and the property

applying those properties we get

Question:7(iii) For some constants a and b, find the derivative of

Given,

Now As we know the quotient rule of derivative,

So applying this rule, we get

Hence

Given,

Now As we know the quotient rule of derivative,

So applying this rule, we get

Hence

Question:9(i) Find the derivative of

Given:

As we know, the property,

and the property

applying that property we get

Question:9(ii) Find the derivative of

Given.

As we know, the property,

and the property

applying that property we get

Question:9(iii) Find the derivative of

Given

As we know, the property,

and the property

applying that property we get

Question:9(iv) Find the derivative of

Given

As we know, the property,

and the property

applying that property we get

Question:9(v) Find the derivative of

Given

As we know, the property,

and the property

applying that property we get

Question:9(vi) Find the derivative of

Given

As we know the quotient rule of derivative:

and the property

So applying this rule, we get

Hence

Given,

f(x)=

Now, As we know, The derivative of any function at x is

Question:11(i) Find the derivative of the following functions:

Given,

f(x)=

Now, As we know the product rule of derivative,

So, applying the rule here,

Question:11(ii) Find the derivative of the following functions:

Given

Now As we know the quotient rule of derivative,

So applying this rule, we get

Question:11 (iii) Find the derivative of the following functions:

Given

As we know the property

Applying the property, we get

Question:11(iv) Find the derivative of the following functions:

Given :

Now As we know the quotient rule of derivative,

So applying this rule, we get

Question:11(v) Find the derivative of the following functions:

Given,

As we know  the property

Applying the property,

Now As we know the quotient rule of derivative,

So applying this rule, we get

Question:11(vi) Find the derivative of the following functions:

Given,

Now as we know the property

So, applying the property,

Question:11(vii) Find the derivative of the following functions:

Given

As we know  the property

Applying this property,

CBSE NCERT solutions for class 11 maths chapter 13 limits and derivatives-Miscellaneous Exercise

Given.

f(x)=-x

Now, As we know, The derivative of any function at x is

Given.

f(x)=

Now, As we know, The derivative of any function at x is

Given.

Now, As we know, The derivative of any function at x is

Given.

Now, As we know, The derivative of any function at x is

Given

f(x)= x + a

As we know, the property,

applying that property we get

Given

As we know, the property,

applying that property we get

Given,

Now,

As we know, the property,

and the property

applying that property we get

Given,

Now, As we know the derivative of any function

Hence, The derivative of f(x) is

Hence Derivative of the function is

.

Given,

Also can be written as

Now, As we know the derivative of any function

Hence, The derivative of f(x) is

Hence Derivative of the function is

Given,

Now, As we know the derivative of any such  function  is given by

Hence, The derivative of f(x) is

Given,

Now, As we know the derivative of any function

Hence, The derivative of f(x) is

Given,

Now, As we know the derivative of any function

Hence, The derivative of f(x) is

Given

As we know, the property,

and the property

applying that property we get

Given

It can also be written as

Now,

As we know, the property,

and the property

applying that property we get

Given

Now, As we know the chain rule of derivative,

And, the property,

Also the property

applying those properties we get,

Given

Now, As we know the chain rule of derivative,

And the Multiplication property of derivative,

And, the property,

Also the property

Applying those properties we get,

Given,

Now, As we know the chain rule of derivative,

Applying this property we get,

Given,

the Multiplication property of derivative,

Applying the property

Hence derivative of the function is .

Given,

Now, As we know the derivative of any function

Hence, The derivative of f(x) is

Given

Also can be written as

which further can be written as

Now,

Given,

which also can be written as

Now,

As we know the derivative of such function

So, The derivative of the function is,

Which can also be written as

.

Given,

Now, As we know the chain rule of derivative,

And, the property,

Applying those properties, we get

Hence Derivative of the given function is

Given Function

Now, As we know the derivative of any function  of this type is:

Hence derivative of the given function will be:

Given,

Now, As we know the derivative of any function

Hence the derivative of the given function is:

Given

Now, As we know, the Multiplication property of derivative,

Hence derivative of the given function is:

Given

Now, As we know the product rule  of derivative,

The derivative of the given function is

Given,

Now As we know the Multiplication property of derivative,(the product rule)

And also the property

Applying those properties we get,

Given,

And the Multiplication property of derivative,

Also the property

Applying those properties we get,

Given,

Now, As we know the derivative of any function

Also the property

Applying those properties,we get

Given,

Now, As we know the derivative of any function

Now, As we know the derivative of any function

Hence the derivative of the given function is

Given

Now, As we know the derivative of any function

Given

Now, As we know  the Multiplication property of derivative,

Also the property

Applying those properties we get,

the derivative of the given function is,

Given,

Now, As we know the derivative of any function

Also chain rule of derivative,

Hence the derivative of the given function is

## NCERT solutions for class 11 mathematics

 chapter-1 NCERT solutions for class 11 maths chapter 1 Sets chapter-2 Solutions of NCERT for class 11 chapter 2 Relations and Functions chapter-3 CBSE NCERT solutions for class 11 chapter 3 Trigonometric Functions chapter-4 NCERT solutions for class 11 chapter 4 Principle of Mathematical Induction chapter-5 Solutions of NCERT for class 11 chapter 5 Complex Numbers and Quadratic equations chapter-6 CBSE NCERT solutions for class 11 maths chapter 6 Linear Inequalities chapter-7 NCERT solutions for class 11 maths chapter 7 Permutation and Combinations chapter-8 Solutions of NCERT for class 11 maths chapter 8 Binomial Theorem chapter-9 CBSE NCERT solutions for class 11 maths chapter 9 Sequences and Series chapter-10 NCERT solutions for class 11 maths chapter 10 Straight Lines chapter-11 Solutions of NCERT for class 11 maths chapter 11 Conic Section chapter-12 CBSE NCERT solutions for class 11 maths chapter 12 Introduction to Three Dimensional Geometry chapter-13