**Exercise 13.1 : Solutions of Questions on Page Number : 301**

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

**Q1 :Evaluate the Given limit: **

**Answer :**

**Q2 :Evaluate the Given limit:**

**Answer :**

**Q3 :Evaluate the Given limit:**

**Answer :**

**Q4 :Evaluate the Given limit:**

**Answer :**

**Q5 :Evaluate the Given limit:**

**Answer :**

**Q6 :Evaluate the Given limit:**

**Answer :
**

Put x + 1 = y so that y Ã¢â€ ’ 1 as x Ã¢â€ ’ 0.

**Q7 :Evaluate the Given limit: **

**Answer :**

At x = 2, the value of the given rational function takes the form

.

**Q8 :Evaluate the Given limit:**

**Answer :**

At x = 2, the value of the given rational function takes the form .

**Q9 :Evaluate the Given limit:**

**Answer** :

**Q10 :Evaluate the Given limit:**

**Answer :
**

At z = 1, the value of the given function takes the form

Put so that z Ã¢â€ ’1 as x Ã¢â€ ’ 1.

**Q11 :Evaluate the Given limit:**

**Answer :**

**Q12 :Evaluate the Given limit**

**Answer :
**

At x = -2, the value of the given function takes the form

.

**Q13 :Evaluate the Given limit:**

**Answer :
**At x = 0, the value of the given function takes the form

**Q14 :Evaluate the Given limit:**

**Answer :
**At x = 0, the value of the given function takes the form

**Q15 :Evaluate the Given limit:**

**Answer :
**It is seen that x Ã¢â€ ’ π ⇒ (π â€“ x) Ã¢â€ ’ 0

**Q16 :Evaluate the given limit:**

**Answer :
**

**Q17 :Evaluate the Given limit:**

**Answer :
**At x = 0, the value of the given function takes the form

Now,

**Q18 :Evaluate the Given limit:**

** Answer :
**

At x = 0, the value of the given function takes the form

Now,

**Q19 :Evaluate the Given limit:**

** Answer :
**

**Q20 :Evaluate the Given limit:**

** Answer :**

At x = 0, the value of the given function takes the form

Now,

**Q21 :Evaluate the Given limit:**

**Answer :**

At x = 0, the value of the given function takes the form ∞ – ∞

Now,

**Q22 :**

**Answer :
**

At , the value of the given function takes the form

Now, put so that

**Q23 :Find f(x) and f(x), where f(x) = **

**Answer :**

The given function is

f(x) =

**Q24 :Find f(x), where f(x) =**

**Answer :**

The given function is

**Q25 :Evaluate f(x), where f(x) =**

**Answer :**

The given function is

f(x) =

**Q26 :Find f(x), where f(x) =**

**Answer :**

The given function is|

**Q27 :Find f(x), where f(x) =**

**Answer :
**The given function is f(x) =

**Q28 :Suppose f(x) = and if f(x) = f(1) what are possible values of a and b?**

** Answer :**

The given function is

Thus, the respective possible values of a and b are 0 and 4.

**Q29 :Let a _{1} , a_{2},……..,a_{n} be fixed real numbers and define a function**

**What is f(x)? For some a ≠ a**

_{1}, a_{2},……..,a_{n}compute f(x).**Answer :**

The given function is

**Q30 :If f(x) =
**

**For what value (s) of a does f(x) exists?**

**Answer :**

The given function is

**When a < 0**

**When a > 0**

Thus, exists for all a ≠ 0.

**Q31 :If the function f(x) satisfies , evaluate **

**Answer :**

**Q32 :If For what integers m and n does and exist?**

**Answer :**

The given function is

Thus, exists if m = n.

Thus, exists for any integral value of m and n.

** Exercise 13.2 : Solutions of Questions on Page Number : 312**

** Q1 :Find the derivative of x ^{2} – 2 at x = 10.**

**Answer :**

Let f(x) = x

^{2}– 2. Accordingly,

Thus, the derivative of x

^{2}– 2 at x = 10 is 20.

**Q2 :Find the derivative of 99x at x = 100.**

**Answer :
**Let f(x) = 99x. Accordingly,

Thus, the derivative of 99x at x = 100 is 99.

**Q3 :Find the derivative of x at x = 1.**

**Answer :**

Letf(x) = x. Accordingly,

Thus, the derivative of x at x = 1 is 1.

**Q4 :Find the derivative of the following functions from first principle.**

**(i) x ^{3} – 27 (ii) (x – 1) (x – 2)**

**(ii) (iv)**

**Answer :**

**(i)**Let f(x) = x

^{3}– 27. Accordingly, from the first principle,

**(ii)**Let f(x) = (x – 1) (x – 2). Accordingly, from the first principle,

**(iii)**Let f(x) = Accordingly, from the first principle,

**(iv)**Let f(x) = Accordingly, from the first principle,

**Q5 :For the function **

**Prove that **

**Answer :**

The given function is

Thus,

**Q6 :Find the derivative of for some fixed real number a.**

**Answer :**

Let

**Q7 :For some constants a and b, find the derivative of**

**(i) (x – a) (x – b) (ii) (ax ^{2} + b)^{2} (iii)**

**Answer :**

**(i)**Let f (x) = (x – a) (x – b)

**(ii)**Let f(x) = (ax

^{2}+ b)

^{2}

**(iii) let f(x) =**

By quotient rule,

**Q8 :Find the derivative of for some constant a.**

**Answer :
**

By quotient rule,

**Q9 :Find the derivative of**

**(i) (ii) (5x ^{3} + 3x – 1) (x – 1)**

**(iii) x**

^{-3}(5 + 3x) (iv) x^{5}(3 – 6x^{-9})**(v) x**

^{-4}(3 – 4x^{-5}) (vi)**Answer :**

**(i)**Let f(x) =

**(ii)**Let f (x) = (5x

^{3}+ 3x – 1) (x – 1)

By Leibnitz product rule,

**(iii)**Let f (x) = x

^{– 3}(5 + 3x)

By Leibnitz product rule,

**(iv)**Let f (x) = x

^{5}(3 – 6

^{x-9})

By Leibnitz product rule,

**(v)**Let f (x) = x

^{-4}(3 – 4

^{x-5})

By Leibnitz product rule,

**(vi)**Let f (x) =

By quotient rule,

**Q10 :Find the derivative of cos x from first principle.**

**Answer :**

Let f (x) = cos x. Accordingly, from the first principle,

**Q11 :Find the derivative of the following functions:**

**(i) sin x cos x (ii) sec x (iii) 5 sec x + 4 cos x**

**(iv) cosec x (v) 3cot x + 5cosec x**

**(vi) 5sin x – 6cos x + 7 (vii) 2tan x – 7sec x**

**Answer :**

**(i)** Let

f (x) = sin x cos x. Accordingly, from the first principle,

**(ii)** Let f (x) = sec x. Accordingly, from the first principle,

**(iii)** Letf (x) = 5 sec x + 4 cos x. Accordingly, from the first principle,

**(iv)** Let f (x) = cosec x. Accordingly, from the first principle,

**(v)** Let f (x) = 3cot x + 5cosec x. Accordingly, from the first principle,

From (1), (2), and (3), we obtain

**(vi)** Let f (x) = 5sin x – 6cos x + 7. Accordingly, from the first principle,

**(vii)** Let f (x) = 2 tan x -7 sec x. Accordingly, from the first principle,

**Exercise Miscellaneous : Solutions of Questions on Page Number : 317**

** Q1 :Find the derivative of the following functions from first principle:**

**(i) -x (ii) (-x) ^{-1} (iii) sin (x + 1)**

**(iv)**

**Answer**:

**(i)**Let f(x) = -x. Accordingly f (x+h)= -(x+h)

By first principle,

**(ii)**Let Accordingly,

By first principle,

**(iii)**Let f(x) = sin (x + 1). Accordingly, f (x+h) =sin (x+h+1)

By first principle,

**(iv)**Let Accordingly,

By first principle,

**Q2 :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): (x + a)**

**Answer :
**Let f(x) = x + a. Accordingly, f(x+h) = x + h + a

By first principle,

**Q3 :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 :
**By Leibnitz product rule,

**Q4 :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): (ax + b) (cx + d) ^{2}**

**Answer :**

Let f (x) =

**(ax + b) (cx + d)**

^{2}By Leibnitz product rule,

**Q5 :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 :**

Let f(x) =

By quotient rule,

**Q6 :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 :
**By quotient rule,

**Q7 :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 :
**Let

By quotient rule,

**Q8 :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 :
**By quotient rule,

**Q9 :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 :
**By quotient rule,

**Q10 :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 :
**

**Q11 :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 :
**

**Q12 :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): (ax + b) ^{n}**

**Answer :**

By first principle,

**Q13 :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): (ax + b) ^{n} (cx + d)^{m}**

**Answer :**

Let f(x) =(ax + b)

^{n}(cx + d)

^{m}

By Leibnitz product rule,

Therefore, from (1), (2), and (3), we obtain

**Q14 :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): sin (x + a)**

**Answer :
**Let f(x) = sin(x+a)

f (h+x) =sin ( h + x+ a)

By first principle,

**Q15 :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): cosec x cot x**

**Answer :
**Let f(x) = cosec x cot x

By Leibnitz product rule,

By first principle,

Now, let f2(x) = cosec x. Accordingly,

By first principle,

From (1), (2), and (3), we obtain

**Q16 :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 :
**Let

By quotient rule,

**Q17 :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 :**

Let

By quotient rule,

**Q18 :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 :**

Let

By quotient rule,

**Q19 :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): sinn x**

**Answer :
**Let y = sin

^{n}x.

Accordingly, for n = 1, y = sin x.

For n = 2, y = sin

^{2}x.

For n = 3, y = sin

^{3}x.

We assert that

Let our assertion be true for n = k.

i.e.,

Thus, our assertion is true for n = k + 1.

Hence, by mathematical induction,

**Q20 :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 :
**By quotient rule,

**Q21 :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 :
**Let

By quotient rule,

By first principle,

From (i) and (ii), we obtain

**Q22 :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): x ^{4} (5 sin x – 3 cos x)**

**Answer :**

Let f(x) =x

^{4}(5 sin x – 3 cos x)

By product rule,

**Q23 :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): (x ^{2} + 1) cos x**

**Answer :**

Let f(x) = (x

^{2}+ 1) cos x

By product rule,

**Q24 :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): (ax ^{2} + sin x) (p + q cos x)**

**Answer :**

Let f(x) = (ax

^{2}+ sin x) (p + q cos x)

By product rule,

**Q25 :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): (x + cosx) (x -tanx)**

**Answer :
**Let f(x) = (x + cosx) (x -tanx)

By product rule,

Let. Accordingly,

By first principle,

Therefore, from (i) and (ii), we obtain

**Q26 :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 :**

Let

By quotient rule,

**Q27 :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 :**

Let

By quotient rule,

**Q28 :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 :**

Let

By first principle,

From (i) and (ii), we obtain

**Q29 :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): (x + sec x) (x – tan x)**

**Answer :
**Let f(x)= (x + sec x) (x – tan x)

By product rule,

From (i), (ii), and (iii), we obtain

**Q30 :****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 :**

Let

By quotient rule,

It can be easily shown that

Therefore,