Limits and Derivatives
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NCERT Solutions for Class 11 Maths Chapter 13
Limits and Derivatives Class 11
Chapter 13 Limits and Derivatives Exercise 13.1, 13.2, miscellaneous Solutions
Exercise 13.1 : Solutions of Questions on Page Number : 301
Q1 :
Evaluate the Given limit:
Answer :
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Q2 :
Evaluate the Given limit:
Answer :
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Q3 :
Evaluate the Given limit:
Answer :
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Q4 :
Evaluate the Given limit:
Answer :
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Q5 :
Evaluate the Given limit:
Answer :
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Q6 :
Evaluate the Given limit:
Answer :
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Put x + 1 = y so that y ΓΆβ€ ’ 1 as x ΓΆβ€ ’ 0.
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Q7 :
Evaluate the Given limit:
Answer :
At x = 2, the value of the given rational function takes the form .
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Q8 :
Evaluate the Given limit:
Answer :
At x = 2, the value of the given rational function takes the form .
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Q9 :
Evaluate the Given limit:
Answer :
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Q10 :
Evaluate the Given limit:
Answer :
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At z = 1, the value of the given
function takes the form .
Put so that z ΓΆβ€ ’1 as x ΓΆβ€ ’ 1.
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Q11 :
Evaluate the Given limit:
Answer :
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Q12 :
Evaluate the Given limit:
Answer :
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At x = –2, the value of the given function takes the form .
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Q13 :
Evaluate the Given limit:
Answer :
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At x = 0, the value of the given
function takes the form .
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Q14 :
Evaluate the Given limit:
Answer :
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At x = 0, the value of the given
function takes the form .
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Q15 :
Evaluate the Given limit:
Answer :
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It is seen that x ΓΆβ€ ’ π ⇒ (π – x) ΓΆβ€ ’ 0
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Q16 :
Evaluate the given
limit:
Answer :
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Q17 :
Evaluate the Given limit:
Answer :
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At x = 0, the value of the given function
takes the form . Now,
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Q18 :
Evaluate the Given limit:
Answer :
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At x = 0, the value of the given
function takes the form .
Now,
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Q19 :
Evaluate the Given limit:
Answer :
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Q20 :
Evaluate the Given limit:
Answer :
At x = 0, the value of the given function takes the form . Now,
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Q21 :
Evaluate the Given limit:
Answer :
At x = 0, the value of the given function
takes the form . Now,
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Q22 :
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Answer :
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At , the value of the given function
takes the form .
Now, put so that .
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Q23 :
Find 
f(x) and f(x),
where f(x) = Answer :
The given function
is
f(x) =
 |
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Q24 :
Find f(x), where f(x) =
Answer :
The given function is
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Q25 :
Evaluate f(x), where f(x) = Answer :
The given function
is
f(x) =
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Q26 :
Find f(x), where f(x) = Answer :
The given function is
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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
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Thus, the respective possible values of a and b are 0 and 4.
Q29 :
Let be fixed real numbers
and define a function
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What is f(x)? For some compute f(x).
Answer :
The given function is .
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Q30 :
If f(x)
= .
For what value (s) of a does f(x) exists?
Answer :
The given function is
When a < 0,
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When a > 0
 |
Thus, exists for all a ≠ 0.
Q31 :
If the function f(x) satisfies , evaluate .
Answer :
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Q32 :
If . For what integers m and n does and exist?
Answer :
The given function is
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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 x2 - 2 at x = 10.
Answer :
Let f(x) = x2 – 2. Accordingly,
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Thus, the derivative of x2 – 2 at x = 10 is 20.
Q2 :
Find the derivative of 99x at x = 100.
Answer :
Let f(x) = 99x. Accordingly,
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Thus, the derivative of 99x at x = 100 is 99.
Q3 :
Find the derivative of x at x = 1.
Answer :
Letf(x) = x. Accordingly,
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Thus, the derivative of x at x = 1 is 1.
Q4 :
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,
Q5 :
For the function
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Prove that
Answer :
The given function is
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Thus,
Q6 :
Find the derivative of for some fixed real number a.
Answer :
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Let
Q7 :
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) (x – b)
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(ii)
 |
Let
(iii)
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By quotient rule,
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Q8 :
Find the derivative of for some constant a.
Answer :
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By quotient rule,
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Q9 :
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
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(ii) Let f (x) = (5x3 + 3x – 1) (x – 1) By Leibnitz product rule,
(iii) Letf (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,
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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) Letf (x) = sin x cos x. Accordingly, from the first principle,
(ii) Letf (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,
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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, By first principle,
(ii) 
Let . Accordingly,
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By first principle,
(iii) Let f(x) = sin (x + 1). Accordingly, 
By first principle,
(iv)
Let . Accordingly,
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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, 
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
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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
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 :
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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 
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
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 
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 = sinn x.
Accordingly, for n = 1, y = sin x.
 |
For n = 2, y = sin2 x.
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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,
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): x4 (5 sin x - 3 cos x)
Answer :
Let
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): (x2 + 1) cos x
Answer :
Let
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): (ax2 + sin x) (p + q cos x)
Answer :
Let
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):
Answer :
Let
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 
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,
