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Limits are considered one of the most important concepts in mathematics, and derivatives are the soul of calculus. Imagine you are climbing a mountain and wonder how steep it is at a certain point. Or, after baking a cake at a high temperature, when you bring it out to room temperature, it will cool down in time, not instantly. You want to know what temperature the cake is at after 30 minutes. Class 11 Maths chapter 12 solutions teach all the relevant concepts for these types of situations, including limits and derivatives. In simpler words, a limit helps us understand the behaviour of a function as it approaches a particular value. On the other hand, a derivative tells us the rate of change of a function. The main purpose of the NCERT class 11 Maths solutions is to strengthen the basics and provide conceptual clarity.
JEE Main Scholarship Test Kit (Class 11): Narayana | Physics Wallah | Aakash | Unacademy
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In class 11 Maths NCERT chapter 9 solutions, students will be introduced to calculus, which is an Important branch of mathematics, as well as study algebra of limits and derivatives with their definitions and certain standard functions. Subject matter experts at Careers360 have curated these NCERT class 11 solutions with step-by-step explanations, ensuring clarity in every step. Students can also explore the following links for syllabus, notes, and PDF: NCERT
Class 11 Maths chapter 12 solutions Exercise: 12.1 Page number: 237-239 Total questions: 32 |
Question 1: Evaluate the following limits
Answer:
Question 2: Evaluate the following limits
Answer:
Below, you can find the solution:
Question 3: Evaluate the following limits
Answer:
The limit
Hence, the answer is
Question 4: Evaluate the following limits
Answer:
The limit
Question 5: Evaluate the following limits
Answer:
The limit
Question 6: Evaluate the following limits
Answer:
The limit
Let's put
Since we have changed the function, its limit will also change,
So
So our function has became
Now, as we know the property
Hence,
Question 7: Evaluate the following limits
Answer:
The limit
Question 8: Evaluate the following limits
Answer:
The limit
At x = 2, both the numerator and denominator become zero, so let's factorise the function
Now we can put the limit directly, so
Question 9: Evaluate the following limits
Answer:
The limit,
Question 10: Evaluate the following limits
Answer:
The limit
Here, on directly putting the limit, both the numerator and the denominator become zero so we factorize the function and then put the limit.
Question 11: Evaluate the following limits
Answer:
The limit:
Since the Denominator is not zero on directly putting the limit, we can directly put the limits, so,
Question 12: Evaluate the following limits
Answer:
Here, since the denominator becomes zero on putting the limit directly, we first simplify the function and then put the limit,
Question 13: Evaluate the following limits
Answer:
The limit
Here, on directly putting the limits, the function becomes
As
Question 14: Evaluate the following limits
Answer:
The limit,
On putting the limit directly, the function takes the zero by zero form. So, we convert it in the form of
Question 15: Evaluate the following limits
Answer:
The limit
Question 16: Evaluate the following limits
Answer:
The limit
The function behaves well on directly putting the limit, so we put the limit directly. So.
Question 17: Evaluate the following limits
Answer:
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
Answer:
The function takes the form zero by zero when we put the limit directly in the function Since theunction consists of the sin function and cos function, we try to make the function in the form of
So,
Question 19: Evaluate the following limits
Answer:
As the function doesn't create any abnormality on putting the limit directly, we can put the limit directly. So,
Question 20: Evaluate the following limits
Answer:
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
Answer:
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
Answer:
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, the limit will also change.
As
So, the function in the new variable becomes,
As we know that property
Question 23: Find
Answer:
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
Answer:
Limit at
Limit at
As we can see Limit at
Question 25: Evaluate
Answer:
The right-hand Limit or Limit at
The left-hand limit or Limit at
Since The left-hand limit and right-hand limit are not equal, The limit of this function at x = 0 does not exist.
Question 26: Evaluate
Answer:
The right-hand Limit or Limit at
The left-hand limit or Limit at
Since The left-hand limit and right-hand limit are not equal, The limit of this function at x = 0 does not exist.
Question 27: Find
Answer:
The right-hand Limit or Limit at
The left-hand limit or Limit at
Since The left-hand limit and right-hand limit are equal, the limit of this function at x = 5 is 0.
Question 28: Suppose
Answer:
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,
Hence, the possible values of a and b are 0 and 4, respectively.
Question 29: Let a1, a2, . . ., an be fixed real numbers and define a function
Answer:
Given,
Now,
Hence,
Now,
Hence
Question 30: If
Answer:
The 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 the Left-hand limit and right-hand limit are not equal, The limit of this function at x = 0 does not exist.
Case 2: When a < 0
The right-hand Limit or Limit at
The left-hand limit or Limit at
Since LHL = RHL, the 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
Answer:
Given
Now,
Question 32: If
Answer:
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.
Class 11 Maths chapter 12 solutions Exercise: 12.2 Page number: 248-249 Total questions: 11 |
Question 1: Find the derivative of
Answer:
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.
Answer:
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.
Answer:
f(x)= 99x
Now, as we know, the derivative of any function at x is
The derivative of f(x) at x = 100:
Question 4 (i): Find the derivative of the following functions from the first principle.
Answer:
Given
f(x)=
Now, as we know, the derivative of any function at x is
Question 4(ii): Find the derivative of the following function from the first principle.
Answer:
f(x)=
Now, as we know, the derivative of any function at x is
Question 4 (iii): Find the derivative of the following functions from the first principle.
Answer:
f(x)=
Now, as we know, the derivative of any function at x is
Question 4 (iv): Find the derivative of the following functions from the first principle.
Answer:
Given:
Now, as we know, the derivative of any function at x is
Question 5: For the function
Answer:
As we know, the property,
Applying that property, we get
Now.
So,
Here
Hence Proved.
Question 6: Find the derivative of
Answer:
Given
As we know, the property,
Applying that property, we get
Question 7(i): For some constants a and b, find the derivative of
Answer:
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
Answer:
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
Answer:
Given,
Now, as we know the quotient rule of derivative,
So,, applying this rule, we get
Hence
Question 8: Find the derivative of
Answer:
Given,
Now, as we know the quotient rule of derivative,
So, applying this rule, we get
Hence
Question 9(i): Find the derivative of
Answer:
Given:
As we know, the property,
And the property
Applying that property, we get
Question 9(ii): Find the derivative of
Answer:
Given.
As we know, the property,
And the property
Applying that property, we get
Question 9(iii): Find the derivative of
Answer:
Given
As we know, the property,
And the property
Applying that property, we get
Question 9(iv): Find the derivative of
Answer:
Given
As we know, the property,
And the property
Applying that property, we get
Question 9(v): Find the derivative of
Answer:
Given
As we know, the property,
And the property
Applying that property, we get
Question 9(vi): Find the derivative of
Answer:
Given
As we know, the quotient rule of the derivative is:
And the property
So, applying this rule, we get
Hence
Question 10: Find the derivative of
Answer:
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:
Answer:
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:
Answer:
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:
Answer:
Given
As we know, the property
Applying the property, we get
Question 11(iv): Find the derivative of the following functions:
Answer:
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:
Answer:
Given,
As we know, the property
Applying the property,
Now, as we know the quotient rule of the derivative,
So, applying this rule, we get
Question 11(vi): Find the derivative of the following functions:
Answer:
Given,
Now, as we know the property
So, applying the property,
Question 11(vii): Find the derivative of the following functions:
Answer:
Given
As we know, the property
Applying this property,
NCERT Limits and Derivatives Class 11 Solutions: Exercise: Miscellaneous Exercise Page Number: 253-254 Total Questions: 30 |
Question 1(i): Find the derivative of the following functions from the first principle: -x
Answer:
Given.
f(x)=-x
Now, as we know, the derivative of any function at x is
Question 1 (ii): Find the derivative of the following functions from the first principle:
Answer:
Given.
f(x)=
Now, as we know, the derivative of any function at x is
Question 1 (iii): Find the derivative of the following functions from the first principle:
Answer:
Given.
Now, as we know, the derivative of any function at x is
Question 1(iv): Find the derivative of the following functions from the first principle:
Answer:
Given.
Now, as we know, the derivative of any function at x is
Answer:
Given
f(x)= x + a
As we know, the property,
Applying that property, we get
Answer:
Given
As we know, the property,
Applying that property, we get
Answer:
Given,
Now,
As we know, the property,
And the property
Applying that property, we get
Answer:
Given,
Now, as we know, the derivative of any function
Hence, the derivative of f(x) is
Hence, the Derivative of the function is
Answer:
Given,
can also be written as
Now, as we know, the derivative of any function
Hence, the derivative of f(x) is
Hence, the Derivative of the function is
Answer:
Given,
Now, as we know, the derivative of any such function is given by
Hence, the derivative of f(x) is
Answer:
Given,
Now, as we know, the derivative of any function
Hence, the derivative of f(x) is
Answer:
Given,
Now, as we know, the derivative of any function
Hence, the derivative of f(x) is
Question 10: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r, and s are fixed non-zero constants and m and n are integers):
Answer:
Given
As we know, the property,
And the property
Applying that property, we get
Question 11: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r, and s are fixed non-zero constants and m and n are integers):
Answer:
Given
It can also be written as
Now,
As we know, the property,
And the property
Applying that property, we get
Question 12: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r, and s are fixed non-zero constants and m and n are integers):
Answer:
Given
Now, ass we know the chain rule of derivative,
And, the property,
Also the property
Applying those properties, we get,
Answer:
Given
Now, ass we know the chain rule of derivative,
And the Multiplication property of the derivative,
And, the property,
Also the property
Applying those properties, we get,
Question 14: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r, and s are fixed non-zero constants and m and n are integers):
Answer:
Given,
Now, ass we know the chain rule of derivative,
Applying this property, we get,
Answer:
Given,
The Multiplication Property of the derivative,
Applying the property
Hence, the derivative of the function is
Answer:
Given,
Now, as we know, the derivative of any function
Hence, the derivative of f(x) is
Answer:
Given
can also be written as
which further can be written as
Now,
Answer:
Given,
which also can be written as
Now,
As we know, the derivative of such a function
So, the derivative of the function is,
This can also be written as
Answer:
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
Question 20: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r, and s are fixed non-zero constants and m and n are integers):
Answer:
Given Function
Now, as we know, the derivative of any function of this type is:
Hence, the derivative of the given function will be:
Answer:
Given,
Now, as we know, the derivative of any function
Hence, the derivative of the given function is:
Answer:
Given
Now, as we know, the Multiplication property of the derivative,
Hence, the derivative of the given function is:
Question 23: Find the derivative of the following functions (it is to be understood that a, b, c, d, p, q, r and s are fixed non-zero constants and m and n are integers):
Answer:
Given
Now, as we know the product rule of derivative,
The derivative of the given function is
Answer:
Given,
Now, as we know, the Multiplication property of the derivative (the product rule)
And also the property
Applying those properties, we get,
Answer:
Given,
And the Multiplication property of the derivative,
Also the property
Applying those properties, we get,
Answer:
Given,
Now, as we know, the derivative of any function
Also the property
Applying those properties, we get
Answer:
Given,
As we know, the derivative of any function
Hence, the derivative of the given function is
Answer:
Given
Noas As we know, the derivative of any function
Answer:
Given
Now, ass we know the Multiplication property of the derivative,
Also the property
Applying those properties, we get,
The derivative of the given function is,
Answer:
Given,
As we know, the derivative of any function
As the chain rule of derivatives,
Hence, the derivative of the given function is
Also, read,
Question:
Solution:
Evaluate the limit:
Rationalise the expression:
Simplify the numerator:
Rewrite the numerator:
Using half-angle formulas:
Cancel common terms:
Evaluate the limit by plugging
Hence, the correct answer is
Given below are the topics discussed in the NCERT Solutions for class 11, chapter 12, Limits and Derivatives:
The limit of a function
Left-Hand Limit:
Right-Hand Limit:
If
Sum Rule:
Constant Rule:
Product Rule:
Quotient Rule:
The derivative of a function
If
Sum Rule:
Difference Rule:
Product Rule:
Quotient Rule:
Here are some approaches that students can use to answer questions related to limits and derivatives.
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Differentiation of Inverse Trigonometric Function (cos/sine/tan) | ✅ | ❌ |
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Differentiation of a Function wrt Another Function and Higher Order derivative of a Function | ✅ | ❌ |
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Non - Removable, Infinite and Oscillatory Type Discontinuity | ✅ | ❌ |
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Application of Extremum in Plane Geometry and Solid geometry | ✅ | ❌ |
Continuity and Discontinuity obtained by Algebraic Operations | ✅ | ❌ |
We at Careers360 compiled all the NCERT class 11 Maths solutions in one place for easy student reference. The following links will allow you to access them
Also Read,
Students can check the following links for more in-depth learning.
Here is the latest NCERT syllabus, that is useful for students before strategising their study plan. Also, it contains links to some reference books which are important for further studies.
Important Topics in NCERT Solutions for Class 11 Maths Chapter 12 (Limits and Derivatives)
This chapter introduces limits and derivatives, forming the foundation of calculus.
1. Limits: Understanding approaching values, left-hand and right-hand limits, and standard limits like:
2. Limit Evaluation Methods:
Direct substitution, factorization, rationalization.
3. Derivatives: First principle definition:
Power rule, sum, product, and quotient rules.
To solve limit problems in Class 11 Maths Chapter 12 (Limits and Derivatives), follow these steps:
1. Substitution: Directly put the given value of
2. Factorization: Factor and cancel common terms if you get
3. Rationalization: Multiply by the conjugate for roots.
4. Trigonometric Limits: Use identities like
5. L'Hôpital's Rule (for indeterminate forms): Differentiate numerator and denominator separately.
Basic Formulas of Limits and Derivatives in Class 11
Limits
1.
2.
3.
4.
5.
Derivatives
1.
2.
3.
4.
5.
To find derivatives using the first principle, we use the definition:
Steps to Find the Derivative Using the First Principle:
1. Substitute
2. Find
3. Simplify the expression and divide by
4. Apply the limit
Example:
Easiest Way to Understand Limits and Derivatives
1. Limits (Approaching a Value)
Think of limits as "getting closer to a value" without necessarily reaching it.
Example:
Use substitution first; if it gives
2. Derivatives (Rate of Change)
The derivative measures how a function changes at a point (slope of a curve).
Formula:
Start with simple functions like
Exam Date:22 July,2025 - 29 July,2025
Exam Date:22 July,2025 - 28 July,2025
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