NCERT Solutions for Class 11 Commerce Maths 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 Commerce students for Maths 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 Commerce Maths 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 Commerce Maths are prepared by experts and are 100% accurate.

#### Page No 301:

#### Question 1:

Evaluate the Given limit:

#### Answer:

#### Page No 301:

#### Question 2:

Evaluate the Given limit:

#### Answer:

#### Page No 301:

#### Question 3:

Evaluate the Given limit:

#### Answer:

#### Page No 301:

#### Question 4:

Evaluate the Given limit:

#### Answer:

#### Page No 301:

#### Question 5:

Evaluate the Given limit:

#### Answer:

#### Page No 301:

#### Question 6:

Evaluate the Given limit:

#### Answer:

Put *x*
+ 1 = *y* so that *y* → 1 as *x* → 0.

#### Page No 301:

#### Question 7:

Evaluate the Given limit:

#### Answer:

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

#### Page No 301:

#### Question 8:

Evaluate the Given limit:

#### Answer:

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

#### Page No 301:

#### Question 9:

Evaluate the Given limit:

#### Answer:

#### Page No 301:

#### 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.

#### Page No 301:

#### Question 11:

Evaluate the Given limit:

#### Answer:

#### Page No 301:

#### Question 12:

Evaluate the Given limit:

#### Answer:

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

#### Page No 301:

#### Question 13:

Evaluate the Given limit:

#### Answer:

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

#### Page No 301:

#### Question 14:

Evaluate the Given limit:

#### Answer:

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

#### Page No 302:

#### Question 15:

Evaluate the Given limit:

#### Answer:

It is seen
that *x* → π ⇒
(π – *x*) → 0

#### Page No 302:

#### Question 16:

Evaluate the given limit:

#### Answer:

#### Page No 302:

#### Question 17:

Evaluate the Given limit:

#### Answer:

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

Now,

#### Page No 302:

#### Question 18:

Evaluate the Given limit:

#### Answer:

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

Now,

#### Page No 302:

#### Question 19:

Evaluate the Given limit:

#### Answer:

#### Page No 302:

#### Question 20:

Evaluate the Given limit:

#### Answer:

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

Now,

#### Page No 302:

#### Question 21:

Evaluate the Given limit:

#### Answer:

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

Now,

#### Page No 302:

#### Question 22:

#### Answer:

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

Now, put so that.

#### Page No 302:

#### Question 23:

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

#### Answer:

The given function is

*f*(*x*) =

#### Page No 302:

#### Question 24:

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

#### Answer:

The given function is

#### Page No 302:

#### Question 25:

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

#### Answer:

The given function is

*f*(*x*)
=

#### Page No 302:

#### Question 26:

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

#### Answer:

The given function is

#### Page No 302:

#### Question 27:

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

#### Answer:

The given
function is *f*(*x*) =.

#### Page No 302:

#### Question 28:

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.

#### Page No 303:

#### Question 29:

Letbe fixed real numbers and define a function

What
is*f*(*x*)?
For some
compute*f*(*x*).

#### Answer:

The given
function is_{.}

#### Page No 303:

#### 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.

#### Page No 303:

#### Question 31:

If the
function *f*(*x*) satisfies,
evaluate.

#### Answer:

#### Page No 303:

#### 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*.

#### Page No 312:

#### Question 1:

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.

#### Page No 312:

#### Question 2:

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

#### Answer:

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

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

#### Page No 312:

#### 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.

#### Page No 312:

#### Question 4:

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. Accordingly, from the first principle,

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

#### Page No 312:

#### Question 5:

For the function

Prove that

#### Answer:

The given function is

Thus,

#### Page No 313:

#### Question 6:

Find the
derivative offor
some fixed real number *a*.

#### Answer:

Let

#### Page No 313:

#### Question 7:

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

(iii)

By quotient rule,

#### Page No 313:

#### Question 8:

Find the
derivative offor
some constant *a*.

#### Answer:

By quotient rule,

#### Page No 313:

#### Question 9:

Find the derivative of

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

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

(v) *x*^{–4}
(3 – 4*x*^{–5}) (vi)

#### Answer:

(i) Let

(ii) Let *f*
(*x*) = (5*x*^{3} + 3*x *– 1) (*x*
– 1)

By Leibnitz product rule,

(iii) Let*
f *(*x*) = *x*^{– 3} (5 + 3*x*)

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,

#### Page No 313:

#### Question 10:

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

#### Answer:

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

#### Page No 313:

#### 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,

#### Page No 317:

#### 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,

#### Page No 317:

#### 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,

#### Page No 317:

#### 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,

#### Page No 317:

#### 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,

#### Page No 317:

#### 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,

#### Page No 317:

#### 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,

#### Page No 317:

#### Question 7:

*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,

#### Page No 317:

#### Question 8:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

By quotient rule,

#### Page No 317:

#### Question 9:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

By quotient rule,

#### Page No 317:

#### Question 10:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

#### Page No 317:

#### Question 11:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

#### Page No 317:

#### 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,

#### Page No 317:

#### 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

#### Page No 317:

#### 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,

#### Page No 317:

#### 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
*f*_{2}(*x*) = cosec *x*. Accordingly,

By first principle,

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

#### Page No 317:

#### Question 16:

*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,

#### Page No 318:

#### Question 17:

*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,

#### Page No 318:

#### Question 18:

*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,

#### Page No 318:

#### 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): sin^{n}* 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,

#### Page No 318:

#### Question 20:

*a*, *b*, *c*, *d*, *p*, q,* r *and* s* are fixed non-zero constants and *m* and *n* are integers):

#### Answer:

By quotient rule,

#### Page No 318:

#### Question 21:

*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

#### Page No 318:

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

#### Answer:

Let

By product rule,

#### Page No 318:

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

#### Answer:

Let

By product rule,

#### Page No 318:

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

#### Answer:

Let

By product rule,

#### Page No 318:

#### Question 25:

*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

#### Page No 318:

#### Question 26:

*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,

#### Page No 318:

#### Question 27:

*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,

#### Page No 318:

#### Question 28:

*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

#### Page No 318:

#### 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

#### Page No 318:

#### Question 30:

*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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