NCERT Solutions for Class 11 Science Math Chapter 13 Limits And Derivatives are provided here with simple step-by-step explanations. These solutions for Limits And Derivatives are extremely popular among Class 11 Science students for Math Limits And Derivatives Solutions come handy for quickly completing your homework and preparing for exams. All questions and answers from the NCERT Book of Class 11 Science Math Chapter 13 are provided here for you for free. You will also love the ad-free experience on Meritnationâ€™s NCERT Solutions. All NCERT Solutions for class Class 11 Science Math are prepared by experts and are 100% accurate.

#### Question 1:

Evaluate the Given limit:

#### Question 2:

Evaluate the Given limit:

#### Question 3:

Evaluate the Given limit:

#### Question 4:

Evaluate the Given limit:

#### Question 5:

Evaluate the Given limit:

#### Question 6:

Evaluate the Given limit:

#### Answer:

Put x + 1 = y so that y → 1 as x → 0.

#### Question 7:

Evaluate the Given limit:

#### Answer:

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

#### Question 8:

Evaluate the Given limit:

#### Answer:

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

#### Question 9:

Evaluate the Given limit:

#### Question 10:

Evaluate the Given limit:

#### Answer:

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

Put so that z →1 as x → 1.

#### Question 11:

Evaluate the Given limit:

#### Question 12:

Evaluate the Given limit:

#### Answer:

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

#### Question 13:

Evaluate the Given limit:

#### Answer:

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

#### Question 14:

Evaluate the Given limit:

#### Answer:

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

#### Question 15:

Evaluate the Given limit:

#### Answer:

It is seen that x → π ⇒ (π – x) → 0

#### Question 16:

Evaluate the given limit:

#### Question 17:

Evaluate the Given limit:

#### Answer:

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

Now,

#### Question 18:

Evaluate the Given limit:

#### Answer:

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

Now,

#### Question 19:

Evaluate the Given limit:

#### Question 20:

Evaluate the Given limit:

#### Answer:

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

Now,

#### Question 21:

Evaluate the Given limit:

#### Answer:

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

Now,

#### Answer:

At, the value of the given function takes the form.

Now, put so that.

#### Question 23:

Find f(x) andf(x), where f(x) =

#### Answer:

The given function is

f(x) =

#### Question 24:

Find f(x), where f(x) =

#### Answer:

The given function is

#### Question 25:

Evaluatef(x), where f(x) =

#### Answer:

The given function is

f(x) =

#### Question 26:

Findf(x), where f(x) =

#### Answer:

The given function is

#### Question 27:

Findf(x), where f(x) =

#### Answer:

The given function is f(x) =.

#### Question 28:

Suppose f(x) = and iff(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.

#### Question 29:

Letbe fixed real numbers and define a function

What isf(x)? For some computef(x).

#### Answer:

The given function is.

#### Question 30:

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.

#### Question 31:

If the function f(x) satisfies, evaluate.

#### Question 32:

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.

#### Question 1:

Find the derivative of x2 – 2 at x = 10.

#### Answer:

Let f(x) = x2 – 2. Accordingly,

Thus, the derivative of x2 – 2 at x = 10 is 20.

#### Question 2:

Find the derivative of 99x at x = 100.

#### Answer:

Let f(x) = 99x. Accordingly,

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

#### Question 3:

Find the derivative of x at x = 1.

#### Answer:

Let f(x) = x. Accordingly,

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

#### Question 4:

Find the derivative of the following functions from first principle.

(i) x3 – 27 (ii) (x – 1) (x – 2)

(ii) (iv)

#### Answer:

(i) Let f(x) = x3 – 27. Accordingly, from the first principle,

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

(iii) Let. Accordingly, from the first principle,

(iv) Let. Accordingly, from the first principle,

For the function

Prove that

#### Answer:

The given function is

Thus,

#### Question 6:

Find the derivative offor some fixed real number a.

Let

#### Question 7:

For some constants a and b, find the derivative of

(i) (x a) (x b) (ii) (ax2 + b)2 (iii)

#### Answer:

(i) Let f (x) = (x a) (xb)

(ii) Let

(iii)

By quotient rule,

#### Question 8:

Find the derivative offor some constant a.

#### Answer:

By quotient rule,

#### Question 9:

Find the derivative of

(i) (ii) (5x3 + 3x – 1) (x – 1)

(iii) x–3 (5 + 3x) (iv) x5 (3 – 6x–9)

(v) x–4 (3 – 4x–5) (vi)

#### Answer:

(i) Let

(ii) Let f (x) = (5x3 + 3x – 1) (x – 1)

By Leibnitz product rule,

(iii) Let f (x) = x– 3 (5 + 3x)

By Leibnitz product rule,

(iv) Let f (x) = x5 (3 – 6x–9)

By Leibnitz product rule,

(v) Let f (x) = x–4 (3 – 4x–5)

By Leibnitz product rule,

(vi) Let f (x) =

By quotient rule,

#### Question 10:

Find the derivative of cos x from first principle.

#### Answer:

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

#### Question 11:

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) Let f (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,

#### Question 1:

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,

By first principle,

(ii) Let. Accordingly,

By first principle,

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

By first principle,

(iv) Let. Accordingly,

By first principle,

#### Question 2:

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,

By first principle,

#### Question 3:

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,

#### Question 4:

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

By Leibnitz product rule,

#### Question 5:

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,

#### Question 6:

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,

#### Question 7:

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,

#### Question 8:

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,

#### Question 9:

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,

#### 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):

#### 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):

#### 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): (ax + b)n

#### Answer:

By first principle,

#### Question 13:

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

By Leibnitz product rule,

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

#### 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): sin (x + a)

#### Answer:

Let

By first principle,

#### Question 15:

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

By Leibnitz product rule,

By first principle,

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

By first principle,

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

#### Question 16:

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,

#### Question 17:

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,

#### Question 18:

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,

#### Question 19:

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 = sinn x.

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

For n = 2, y = sin2 x.

For n = 3, y = sin3 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,

#### 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:

By quotient rule,

#### Question 21:

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

#### Question 22:

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): x4 (5 sin x – 3 cos x)

Let

By product rule,

#### 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): (x2 + 1) cos x

Let

By product rule,

#### Question 24:

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): (ax2 + sin x) (p + q cos x)

Let

By product rule,

#### Question 25:

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 product rule,

Let. Accordingly,

By first principle,

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

#### Question 26:

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,

#### Question 27:

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,

#### Question 28:

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

#### Question 29:

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

By product rule,

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

#### Question 30:

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,

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