Important Questions for Class 11 Maths Chapter 13 - Limits and Derivatives

Get the important questions for class 11 Maths Chapter 13 – Limits and Derivatives here. The important questions are given here will help you in the preparation for the annual examination. Go through the problems and clear your doubts while solving the problems. Students can get an idea about the type of questions asked in the examination after solving these questions. Here, the questions are based on the NCERT textbook and as per the syllabus of the CBSE board. The problems include all types of questions such as 1 mark, 2 marks, 4 marks, 6 marks. Practice the problems provided here to achieve a good score in class 11 Maths final examination. Also, get all the chapters important questions for Maths here.

Class 11 Maths Chapter 13 – Limits and Derivatives includes the following important concepts such as:

• Limits
• Derivatives
• Limits of the trigonometric functions
• Algebra of the derivative of the function

Also, Check:

• Important 1 Mark Questions for CBSE Class 11 Maths
• Important 4 Marks Questions for CBSE Class 11 Maths
• Important 6 Marks Questions for CBSE Class 11 Maths

Class 11 Chapter 13 – Limits and Derivatives Important Questions with Solutions

Practice the following important questions in class 11 Maths Limits and Derivatives that should help you to solve the problems faster with accuracy.

Question 1:

Find the derivative of the function x²cos x.

Solution:

Given function is x²cos x

Let y = x²cos x

Differentiate with respect to x on both sides.

Then, we get:

dy/dx = (d/dx)x²cos x

Now, using the formula, we can write the above form as:

dy/dx = x² (d/dx) cos x + cos x (d/dx)x²

Now, differentiate the function:

dy/dx = x² (-sin x) + cos x (2x)

Now, rearrange the terms, we will get:

dy/dx = 2x cos x – x² sin x

Question 2:

Find the positive integer “n” so that lim_{x → 3}[(xⁿ– 3ⁿ)/(x – 3)] = 108.

Solution:

Given limit: lim_{x → 3}[(xⁿ– 3ⁿ)/(x – 3)] = 108

Now, we have:

lim_{x → 3}[(xⁿ– 3ⁿ)/(x-3)] = n(3)^n-1

n(3)^n-1 = 108

Now, this can be written as:

n(3)^n-1 = 4 (27) = 4(3)^4-1

Therefore, by comparing the exponents in the above equation, we get:

n = 4

Therefore, the value of positive integer “n” is 4.

Question 3:

Find the derivative of f(x) = x³ using the first principle.

Solution:

By definition,

f’(x) = lim_{h→ 0} [f(x+h)-f(x)]/h

Now, substitute f(x)=x³ in the above equation:

f’(x) = lim_{h→ 0} [(x+h)³-x³]/h

f’(x) = lim_{h→ 0} (x³+h³+3xh(x+h)-x³)/h

f’(x) = lim_{h→ 0} (h²+3x(x+h))

Substitute h = 0, we get:

f’(x) = 3x²

Therefore, the derivative of the function f’(x) = x³ is 3x².

Question 4:

Determine the derivative of cosx/(1+sin x).

Solution:

Given function: cosx/(1+sin x)

Let y = cosx/(1+sin x)

Now, differentiate the function with respect to “x”, we get

dy/dx = (d/dx) (cos x/(1+sin x))

Now, use the u/v formula in the above form, we get

dy/dx = [(1+sin x)(-sin x) – (cos x)(cos x)]/(1+sin x)²

dy/dx = (-sin x – sin²x-cos²x)/(1+sin x)²

Now, take (-) outside from the numerator, we get:

dy/dx = -(sin x + sin^2.x + cos²x)/(1+sin x)²

We know that sin^2.x + cos²x = 1

By substituting this, we can get:

dy/dx = -(1+sin x)/(1+sin x)²

Cancel out (1+sin x) from both numerator and denominator, we get:

dy/dx = -1/(1+sin x)

Therefore, the derivative of cosx/(1+sin x) is -1/(1+sin x).

Question 5:

lim_{x→ 0} |x|/x is equal to:

(a)1 (b)-1 (c)0 (d)does not exists

Solution:

A correct answer is an option (d)

Explanation:

The limit mentioned here is x→0

It has two possibilities:

Case 1: x→0⁺

Now, substitute the limit in the given function:

lim_{x→ 0+} |x|/x = x/x = 1

Case 2: x→0^–

Now, substitute the limit in the given function:

lim_{x→ 0-} |x|/x = -x/x = -1

Hence, the result for both cases varies, the solution is an option (D)

Question 6:

Evaluate the derivative of f(x) = sin²x using Leibnitz product rule.

Solution:

Given function: f(x) = sin²x

Let y= sin²x

Now, by using Leibnitz product rule, we can write it as:

dy/dx = (d/dx) sin²x

Sin²x can be written as (sin x)(sin x)

Now, it becomes:

dy/dx = (d/dx) (sin x)(sin x)

dy/dx = (sin x)’(sin x) + (sin x)(sin x)’

dy/dx = cos x sin x + sin x cos x

dy/dx = 2 sin x cos x

dy/dx = sin 2x

Therefore, the derivative of the function sin²x is sin 2x.

Practice Problems for Class 11 Maths Chapter 13 – Limits and Derivatives

Solve chapter 13 limits and derivatives important problems given below:

1. Evaluate: lim_{x → 0} [(sin²2x)/(sin²4x)]
2. Differentiate the function with respect to x: (ax² + cot x)(p+q cos x)
3. Show that the lim_{x → 0} [(|x-4|)/(x-4)] does not exists
4. Evaluate the following:
   lim_{y → 0} [(x+y)sec(x+y)-x sec x]/y
5. Differentiate 1/(ax²+bx+c) with respect to x.
6. Evaluate the derivative of 99x at x=100
7. Find the derivative of the following trigonometric functions:
   (i) 2 tan x – 7 sec x
   (ii) sin x cos x
   (iii) 5 sec x + 4 cos x
8. Differentiate the function: cos (x²+1).
9. Differentiate x² sin x + cos 2 x.
10. Differentiate (2x – 7)² (3x+5)³.

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