# MATHEMATICS XI

## Solutions Chapter 13 – Limits and Derivatives

#### 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:

Put x + 1 = y so that y → 1 as x → 0.

#### Question 7:

Evaluate the Given limit:

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

#### Question 8:

Evaluate the Given limit:

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:

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:

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

#### Question 13:

Evaluate the Given limit:

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

#### Question 14:

Evaluate the Given limit:

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

#### Question 15:

Evaluate the Given limit:

It is seen that x → π ⇒ (π – x) → 0

#### Question 16:

Evaluate the given limit:

#### Question 17:

Evaluate the Given limit:

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

Now,

#### Question 18:

Evaluate the Given limit:

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:

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

Now,

#### Question 21:

Evaluate the Given limit:

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

Now,

#### Question 22:

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

Now, put so that.

#### Question 23:

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

The given function is

f(x) =

#### Question 24:

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

The given function is

#### Question 25:

Evaluatef(x), where f(x) =

The given function is

f(x) =

#### Question 26:

Findf(x), where f(x) =

The given function is

#### Question 27:

Findf(x), where f(x) =

The given function is f(x) =.

#### Question 28:

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

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).

The given function is.

#### Question 30:

If f(x) =.

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

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?

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.

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.

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.

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)

(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,

#### Question 5:

For the function

Prove that

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)

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

(ii) Let

(iii)

By quotient rule,

#### Question 8:

Find the derivative offor some constant a.

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)

(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.

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

(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)

(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)

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):

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

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):

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):

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):

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):

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):

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

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

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)

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

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):

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):

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):

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

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):

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):

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):

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):

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):

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):

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)

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):