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Mathematics
Limits and Derivatives
NCERT Solutions
NCERT Solutions
Limits and Derivatives
72 Solutions
Exercise:
All Exercises
EXERCISE 12.1
EXERCISE 12.2
Miscellaneous Exercise on Chapter 12
Q1
EXERCISE 12.1
Evaluate the following limits in Exercises 1 to 22.
lim
x
→
3
x
+
3
\lim_{x \rightarrow 3} x+3
lim
x
→
3
x
+
3
Q2
EXERCISE 12.1
lim
x
→
π
(
x
−
22
7
)
\lim_{x \rightarrow \pi} \left(x-\frac{22}{7}\right)
lim
x
→
π
(
x
−
7
22
)
Q3
EXERCISE 12.1
lim
r
→
1
π
r
2
\lim_{r \rightarrow 1} \pi r^{2}
lim
r
→
1
π
r
2
Q4
EXERCISE 12.1
lim
x
→
4
4
x
+
3
x
−
2
\lim_{x \rightarrow 4} \frac{4x+3}{x-2}
lim
x
→
4
x
−
2
4
x
+
3
Q5
EXERCISE 12.1
lim
x
→
−
1
x
10
+
x
5
+
1
x
−
1
\lim_{x \rightarrow -1} \frac{x^{10}+x^{5}+1}{x-1}
lim
x
→
−
1
x
−
1
x
10
+
x
5
+
1
Q6
EXERCISE 12.1
lim
x
→
0
(
x
+
1
)
5
−
1
x
\lim_{x \rightarrow 0} \frac{(x+1)^{5}-1}{x}
lim
x
→
0
x
(
x
+
1
)
5
−
1
Q7
EXERCISE 12.1
lim
x
→
2
3
x
2
−
x
−
10
x
2
−
4
\lim_{x \rightarrow 2} \frac{3x^2-x-10}{x^2-4}
lim
x
→
2
x
2
−
4
3
x
2
−
x
−
10
Q8
EXERCISE 12.1
lim
x
→
3
x
4
−
81
2
x
2
−
5
x
−
3
\lim_{x \rightarrow 3} \frac{x^4-81}{2x^2-5x-3}
lim
x
→
3
2
x
2
−
5
x
−
3
x
4
−
81
Q9
EXERCISE 12.1
lim
x
→
0
a
x
+
b
c
x
+
1
\lim_{x \rightarrow 0} \frac{ax+b}{cx+1}
lim
x
→
0
c
x
+
1
a
x
+
b
Q10
EXERCISE 12.1
lim
z
→
1
z
1
3
−
1
z
1
6
−
1
\lim_{z \rightarrow 1} \frac{z^{\frac{1}{3}}-1}{z^{\frac{1}{6}}-1}
lim
z
→
1
z
6
1
−
1
z
3
1
−
1
Q11
EXERCISE 12.1
lim
x
→
1
a
x
2
+
b
x
+
c
c
x
2
+
b
x
+
a
,
a
+
b
+
c
≠
0
\lim_{x \rightarrow 1} \frac{ax^2+bx+c}{cx^2+bx+a}, a+b+c \neq 0
lim
x
→
1
c
x
2
+
b
x
+
a
a
x
2
+
b
x
+
c
,
a
+
b
+
c
=
0
Q12
EXERCISE 12.1
lim
x
→
−
2
1
x
+
1
2
x
+
2
\lim_{x \rightarrow -2} \frac{\frac{1}{x}+\frac{1}{2}}{x+2}
lim
x
→
−
2
x
+
2
x
1
+
2
1
Q13
EXERCISE 12.1
lim
x
→
0
sin
a
x
b
x
\lim_{x \rightarrow 0} \frac{\sin ax}{bx}
lim
x
→
0
b
x
s
i
n
a
x
Q14
EXERCISE 12.1
lim
x
→
0
sin
a
x
sin
b
x
,
a
,
b
≠
0
\lim_{x \rightarrow 0} \frac{\sin ax}{\sin bx}, a, b \neq 0
lim
x
→
0
s
i
n
b
x
s
i
n
a
x
,
a
,
b
=
0
Q15
EXERCISE 12.1
lim
x
→
π
sin
(
π
−
x
)
π
(
π
−
x
)
\lim_{x \rightarrow \pi} \frac{\sin (\pi-x)}{\pi(\pi-x)}
lim
x
→
π
π
(
π
−
x
)
s
i
n
(
π
−
x
)
Q16
EXERCISE 12.1
lim
x
→
0
cos
x
π
−
x
\lim_{x \rightarrow 0} \frac{\cos x}{\pi-x}
lim
x
→
0
π
−
x
c
o
s
x
Q17
EXERCISE 12.1
lim
x
→
0
cos
2
x
−
1
cos
x
−
1
\lim_{x \rightarrow 0} \frac{\cos 2x-1}{\cos x-1}
lim
x
→
0
c
o
s
x
−
1
c
o
s
2
x
−
1
Q18
EXERCISE 12.1
lim
x
→
0
a
x
+
x
cos
x
b
sin
x
\lim_{x \rightarrow 0} \frac{ax+x \cos x}{b \sin x}
lim
x
→
0
b
s
i
n
x
a
x
+
x
c
o
s
x
Q19
EXERCISE 12.1
lim
x
→
0
x
sec
x
\lim_{x \rightarrow 0} x \sec x
lim
x
→
0
x
sec
x
Q20
EXERCISE 12.1
lim
x
→
0
sin
a
x
+
b
x
a
x
+
sin
b
x
a
,
b
,
a
+
b
≠
0
\lim_{x \rightarrow 0} \frac{\sin ax+bx}{ax+\sin bx} a, b, a+b \neq 0
lim
x
→
0
a
x
+
s
i
n
b
x
s
i
n
a
x
+
b
x
a
,
b
,
a
+
b
=
0
Q21
EXERCISE 12.1
lim
x
→
0
(
csc
x
−
cot
x
)
\lim_{x \rightarrow 0} (\csc x - \cot x)
lim
x
→
0
(
csc
x
−
cot
x
)
Q22
EXERCISE 12.1
lim
x
→
π
2
tan
2
x
x
−
π
2
\lim_{x \rightarrow \frac{\pi}{2}} \frac{\tan 2x}{x-\frac{\pi}{2}}
lim
x
→
2
π
x
−
2
π
t
a
n
2
x
Q23
EXERCISE 12.1
Find
lim
x
→
0
f
(
x
)
\lim_{x \rightarrow 0} f(x)
lim
x
→
0
f
(
x
)
and
lim
x
→
1
f
(
x
)
\lim_{x \rightarrow 1} f(x)
lim
x
→
1
f
(
x
)
, where
f
(
x
)
=
{
2
x
+
3
,
x
≤
0
3
(
x
+
1
)
,
x
>
0
f(x)=\begin{cases}2 x+3, & x \leq 0 \ 3(x+1), & x>0\end{cases}
f
(
x
)
=
{
2
x
+
3
,
x
≤
0
3
(
x
+
1
)
,
x
>
0
Q24
EXERCISE 12.1
Find
lim
x
→
1
f
(
x
)
\lim_{x \rightarrow 1} f(x)
lim
x
→
1
f
(
x
)
, where
f
(
x
)
=
{
x
2
−
1
,
x
≤
1
−
x
2
−
1
,
x
>
1
f(x)= \begin{cases}x^{2}-1, & x \leq 1 \ -x^{2}-1, & x>1\end{cases}
f
(
x
)
=
{
x
2
−
1
,
x
≤
1
−
x
2
−
1
,
x
>
1
Q25
EXERCISE 12.1
Evaluate
lim
x
→
0
f
(
x
)
\lim_{x \rightarrow 0} f(x)
lim
x
→
0
f
(
x
)
, where
f
(
x
)
=
{
∣
x
∣
x
,
x
≠
0
0
,
x
=
0
f(x)= \begin{cases}\frac{|x|}{x}, & x \neq 0 \ 0, & x=0\end{cases}
f
(
x
)
=
{
x
∣
x
∣
,
x
=
0
0
,
x
=
0
Q26
EXERCISE 12.1
Find
lim
x
→
0
f
(
x
)
\lim_{x \rightarrow 0} f(x)
lim
x
→
0
f
(
x
)
, where
f
(
x
)
=
{
x
∣
x
∣
,
x
≠
0
0
,
x
=
0
f(x)=\begin{cases}\frac{x}{|x|}, & x \neq 0 \ 0, & x=0\end{cases}
f
(
x
)
=
{
∣
x
∣
x
,
x
=
0
0
,
x
=
0
Q27
EXERCISE 12.1
Find
lim
x
→
5
f
(
x
)
\lim_{x \rightarrow 5} f(x)
lim
x
→
5
f
(
x
)
, where
f
(
x
)
=
∣
x
∣
−
5
f(x)=|x|-5
f
(
x
)
=
∣
x
∣
−
5
Q28
EXERCISE 12.1
Suppose
f
(
x
)
=
{
a
+
b
x
,
x
<
1
4
,
x
=
1
b
−
a
x
,
x
>
1
f(x)= \begin{cases}a+b x, & x<1 \ 4, & x=1 \ b-a x, & x>1\end{cases}
f
(
x
)
=
{
a
+
b
x
,
x
<
1
4
,
x
=
1
b
−
a
x
,
x
>
1
and if
lim
x
→
1
f
(
x
)
=
f
(
1
)
\lim_{x \rightarrow 1} f(x)=f(1)
lim
x
→
1
f
(
x
)
=
f
(
1
)
what are possible values of
a
a
a
and
b
b
b
?
Q29
EXERCISE 12.1
Let
a
1
,
a
2
,
…
,
a
n
a_{1}, a_{2}, \ldots, a_{n}
a
1
,
a
2
,
…
,
a
n
be fixed real numbers and define a function
f
(
x
)
=
(
x
−
a
1
)
(
x
−
a
2
)
…
(
x
−
a
n
)
f(x)=\left(x-a_{1}\right)\left(x-a_{2}\right) \ldots\left(x-a_{n}\right)
f
(
x
)
=
(
x
−
a
1
)
(
x
−
a
2
)
…
(
x
−
a
n
)
What is
lim
x
→
a
1
f
(
x
)
\lim _{x \rightarrow a_{1}} f(x)
lim
x
→
a
1
f
(
x
)
? For some
a
≠
a
1
,
a
2
,
…
,
a
n
a \neq a_{1}, a_{2}, \ldots, a_{n}
a
=
a
1
,
a
2
,
…
,
a
n
, compute
lim
x
→
a
f
(
x
)
\lim _{x \rightarrow a} f(x)
lim
x
→
a
f
(
x
)
Q30
EXERCISE 12.1
If
f
(
x
)
=
{
∣
x
∣
+
1
,
x
<
0
0
,
x
=
0
∣
x
∣
−
1
,
x
>
0
f(x)=\begin{cases}|x|+1, & x<0 \ 0, & x=0 \ |x|-1, & x>0\end{cases}
f
(
x
)
=
{
∣
x
∣
+
1
,
x
<
0
0
,
x
=
0
∣
x
∣
−
1
,
x
>
0
. For what value (s) of
a
a
a
does
lim
x
→
a
f
(
x
)
\lim_{x \rightarrow a} f(x)
lim
x
→
a
f
(
x
)
exists?
Q31
EXERCISE 12.1
If the function
f
(
x
)
f(x)
f
(
x
)
satisfies
lim
x
→
1
f
(
x
)
−
2
x
2
−
1
=
π
\lim_{x \rightarrow 1} \frac{f(x)-2}{x^{2}-1}=\pi
lim
x
→
1
x
2
−
1
f
(
x
)
−
2
=
π
, evaluate
lim
x
→
1
f
(
x
)
\lim_{x \rightarrow 1} f(x)
lim
x
→
1
f
(
x
)
Q32
EXERCISE 12.1
If
f
(
x
)
=
{
m
x
2
+
n
,
x
<
0
n
x
+
m
,
0
≤
x
≤
1
n
x
3
+
m
,
x
>
1
f(x)=\begin{cases}m x^{2}+n, & x<0 \ n x+m, & 0 \leq x \leq 1 \ n x^{3}+m, & x>1\end{cases}
f
(
x
)
=
{
m
x
2
+
n
,
x
<
0
n
x
+
m
,
0
≤
x
≤
1
n
x
3
+
m
,
x
>
1
. For what integers
m
m
m
and
n
n
n
does both
lim
x
→
0
f
(
x
)
\lim_{x \rightarrow 0} f(x)
lim
x
→
0
f
(
x
)
and
lim
x
→
1
f
(
x
)
\lim_{x \rightarrow 1} f(x)
lim
x
→
1
f
(
x
)
exist?
Q1
EXERCISE 12.2
Find the derivative of
x
2
−
2
x^2-2
x
2
−
2
at
x
=
10
x=10
x
=
10
.
Q2
EXERCISE 12.2
Find the derivative of
x
x
x
at
x
=
1
x=1
x
=
1
.
Q3
EXERCISE 12.2
Find the derivative of
99
x
99x
99
x
at
x
=
100
x=100
x
=
100
.
Q4
EXERCISE 12.2
Find the derivative of the following functions from first principle.
(i)
x
3
−
27
x^3-27
x
3
−
27
(ii)
(
x
−
1
)
(
x
−
2
)
(x-1)(x-2)
(
x
−
1
)
(
x
−
2
)
(iii)
1
x
2
\frac{1}{x^2}
x
2
1
(iv)
x
+
1
x
−
1
\frac{x+1}{x-1}
x
−
1
x
+
1
Q5
EXERCISE 12.2
For the function
f
(
x
)
=
x
100
100
+
x
99
99
+
…
+
x
2
2
+
x
+
1
f(x)=\frac{x^{100}}{100}+\frac{x^{99}}{99}+\ldots+\frac{x^{2}}{2}+x+1
f
(
x
)
=
100
x
100
+
99
x
99
+
…
+
2
x
2
+
x
+
1
Prove that
f
′
(
1
)
=
100
f
′
(
0
)
f^{\prime}(1)=100 f^{\prime}(0)
f
′
(
1
)
=
100
f
′
(
0
)
.
Q6
EXERCISE 12.2
Find the derivative of
x
n
+
a
x
n
−
1
+
a
2
x
n
−
2
+
…
+
a
n
−
1
x
+
a
n
x^{n}+a x^{n-1}+a^{2} x^{n-2}+\ldots+a^{n-1} x+a^{n}
x
n
+
a
x
n
−
1
+
a
2
x
n
−
2
+
…
+
a
n
−
1
x
+
a
n
for some fixed real number
a
a
a
.
Q7
EXERCISE 12.2
For some constants
a
a
a
and
b
b
b
, find the derivative of
(i)
(
x
−
a
)
(
x
−
b
)
(x-a)(x-b)
(
x
−
a
)
(
x
−
b
)
(ii)
(
a
x
2
+
b
)
2
(ax^2+b)^2
(
a
x
2
+
b
)
2
(iii)
x
−
a
x
−
b
\frac{x-a}{x-b}
x
−
b
x
−
a
Q8
EXERCISE 12.2
Find the derivative of
x
n
−
a
n
x
−
a
\frac{x^{n}-a^{n}}{x-a}
x
−
a
x
n
−
a
n
for some constant
a
a
a
.
Q9
EXERCISE 12.2
Find the derivative of
(i)
2
x
−
3
4
2x - \frac{3}{4}
2
x
−
4
3
(ii)
(
5
x
3
+
3
x
−
1
)
(
x
−
1
)
(5x^3+3x-1)(x-1)
(
5
x
3
+
3
x
−
1
)
(
x
−
1
)
(iii)
x
−
3
(
5
+
3
x
)
x^{-3}(5+3x)
x
−
3
(
5
+
3
x
)
(iv)
x
5
(
3
−
6
x
−
9
)
x^5(3-6x^{-9})
x
5
(
3
−
6
x
−
9
)
(v)
x
−
4
(
3
−
4
x
−
5
)
x^{-4}(3-4x^{-5})
x
−
4
(
3
−
4
x
−
5
)
(vi)
2
x
+
1
−
x
2
3
x
−
1
\frac{2}{x+1} - \frac{x^2}{3x-1}
x
+
1
2
−
3
x
−
1
x
2
Q10
EXERCISE 12.2
Find the derivative of
cos
x
\cos x
cos
x
from first principle.
Q11
EXERCISE 12.2
Find the derivative of the following functions:
(i)
sin
x
cos
x
\sin x \cos x
sin
x
cos
x
(ii)
sec
x
\sec x
sec
x
(iii)
5
sec
x
+
4
cos
x
5 \sec x+4 \cos x
5
sec
x
+
4
cos
x
(iv)
csc
x
\csc x
csc
x
(v)
3
cot
x
+
5
csc
x
3 \cot x+5 \csc x
3
cot
x
+
5
csc
x
(vi)
5
sin
x
−
6
cos
x
+
7
5 \sin x-6 \cos x+7
5
sin
x
−
6
cos
x
+
7
(vii)
2
tan
x
−
7
sec
x
2 \tan x-7 \sec x
2
tan
x
−
7
sec
x
Q1
Miscellaneous Exercise on Chapter 12
Find the derivative of the following functions from first principle:
(i)
−
x
-x
−
x
(ii)
(
−
x
)
−
1
(-x)^{-1}
(
−
x
)
−
1
(iii)
sin
(
x
+
1
)
\sin (x+1)
sin
(
x
+
1
)
(iv)
cos
(
x
−
π
8
)
\cos \left(x-\frac{\pi}{8}\right)
cos
(
x
−
8
π
)
Q2
Miscellaneous Exercise on Chapter 12
Find the derivative of the following functions (it is to be understood that
a
,
b
,
c
,
d
,
p
,
q
,
r
a, b, c, d, p, q, r
a
,
b
,
c
,
d
,
p
,
q
,
r
and
s
s
s
are fixed non-zero constants and
m
m
m
and
n
n
n
are integers): 2.
(
x
+
a
)
(x+a)
(
x
+
a
)
Q3
Miscellaneous Exercise on Chapter 12
(
p
x
+
q
)
(
r
x
+
s
)
(p x+q)\left(\frac{r}{x}+s\right)
(
p
x
+
q
)
(
x
r
+
s
)
Q4
Miscellaneous Exercise on Chapter 12
(
a
x
+
b
)
(
c
x
+
d
)
2
(a x+b)(c x+d)^{2}
(
a
x
+
b
)
(
c
x
+
d
)
2
Q5
Miscellaneous Exercise on Chapter 12
a
x
+
b
c
x
+
d
\frac{a x+b}{c x+d}
c
x
+
d
a
x
+
b
Q6
Miscellaneous Exercise on Chapter 12
1
+
1
x
1
−
1
x
\frac{1+\frac{1}{x}}{1-\frac{1}{x}}
1
−
x
1
1
+
x
1
Q7
Miscellaneous Exercise on Chapter 12
1
a
x
2
+
b
x
+
c
\frac{1}{a x^{2}+b x+c}
a
x
2
+
b
x
+
c
1
Q8
Miscellaneous Exercise on Chapter 12
a
x
+
b
p
x
2
+
q
x
+
r
\frac{a x+b}{p x^{2}+q x+r}
p
x
2
+
q
x
+
r
a
x
+
b
Q9
Miscellaneous Exercise on Chapter 12
p
x
2
+
q
x
+
r
a
x
+
b
\frac{p x^{2}+q x+r}{a x+b}
a
x
+
b
p
x
2
+
q
x
+
r
Q10
Miscellaneous Exercise on Chapter 12
a
x
4
−
b
x
2
+
cos
x
\frac{a}{x^{4}}-\frac{b}{x^{2}}+\cos x
x
4
a
−
x
2
b
+
cos
x
Q11
Miscellaneous Exercise on Chapter 12
4
x
−
2
4 \sqrt{x}-2
4
x
−
2
Q12
Miscellaneous Exercise on Chapter 12
(
a
x
+
b
)
n
(a x+b)^{n}
(
a
x
+
b
)
n
Q13
Miscellaneous Exercise on Chapter 12
(
a
x
+
b
)
n
(
c
x
+
d
)
m
(a x+b)^{n}(c x+d)^{m}
(
a
x
+
b
)
n
(
c
x
+
d
)
m
Q14
Miscellaneous Exercise on Chapter 12
sin
(
x
+
a
)
\sin (x+a)
sin
(
x
+
a
)
Q15
Miscellaneous Exercise on Chapter 12
csc
x
cot
x
\csc x \cot x
csc
x
cot
x
Q16
Miscellaneous Exercise on Chapter 12
cos
x
1
+
sin
x
\frac{\cos x}{1+\sin x}
1
+
s
i
n
x
c
o
s
x
Q17
Miscellaneous Exercise on Chapter 12
sin
x
+
cos
x
sin
x
−
cos
x
\frac{\sin x+\cos x}{\sin x-\cos x}
s
i
n
x
−
c
o
s
x
s
i
n
x
+
c
o
s
x
Q18
Miscellaneous Exercise on Chapter 12
sec
x
−
1
sec
x
+
1
\frac{\sec x-1}{\sec x+1}
s
e
c
x
+
1
s
e
c
x
−
1
Q19
Miscellaneous Exercise on Chapter 12
sin
n
x
\sin ^{n} x
sin
n
x
Q20
Miscellaneous Exercise on Chapter 12
a
+
b
sin
x
c
+
d
cos
x
\frac{a+b \sin x}{c+d \cos x}
c
+
d
c
o
s
x
a
+
b
s
i
n
x
Q21
Miscellaneous Exercise on Chapter 12
sin
(
x
+
a
)
cos
x
\frac{\sin (x+a)}{\cos x}
c
o
s
x
s
i
n
(
x
+
a
)
Q22
Miscellaneous Exercise on Chapter 12
x
4
(
5
sin
x
−
3
cos
x
)
x^{4}(5 \sin x-3 \cos x)
x
4
(
5
sin
x
−
3
cos
x
)
Q23
Miscellaneous Exercise on Chapter 12
(
x
2
+
1
)
cos
x
\left(x^{2}+1\right) \cos x
(
x
2
+
1
)
cos
x
Q24
Miscellaneous Exercise on Chapter 12
(
a
x
2
+
sin
x
)
(
p
+
q
cos
x
)
\left(a x^{2}+\sin x\right)(p+q \cos x)
(
a
x
2
+
sin
x
)
(
p
+
q
cos
x
)
Q25
Miscellaneous Exercise on Chapter 12
(
x
+
cos
x
)
(
x
−
tan
x
)
(x+\cos x)(x-\tan x)
(
x
+
cos
x
)
(
x
−
tan
x
)
Q26
Miscellaneous Exercise on Chapter 12
4
x
+
5
sin
x
3
x
+
7
cos
x
\frac{4 x+5 \sin x}{3 x+7 \cos x}
3
x
+
7
c
o
s
x
4
x
+
5
s
i
n
x
Q27
Miscellaneous Exercise on Chapter 12
x
2
cos
(
π
4
)
sin
x
\frac{x^{2} \cos \left(\frac{\pi}{4}\right)}{\sin x}
s
i
n
x
x
2
c
o
s
(
4
π
)
Q29
Miscellaneous Exercise on Chapter 12
(
x
+
sec
x
)
(
x
−
tan
x
)
(x+\sec x)(x-\tan x)
(
x
+
sec
x
)
(
x
−
tan
x
)
Q30
Miscellaneous Exercise on Chapter 12
x
sin
n
x
\frac{x}{\sin ^{n} x}
s
i
n
n
x
x
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