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Week 7 Mathematics 1 Questions | Prasnya
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46Q
01
The value of
lim
x
→
0
(
1
+
x
)
1
/
2
−
(
1
−
x
)
1
/
2
x
\displaystyle \lim_{x\to 0}\frac{(1+x)^{1/2}-(1-x)^{1/2}}{x}
x
→
0
lim
x
(
1
+
x
)
1/2
−
(
1
−
x
)
1/2
is:
Single correct
EASY
3 marks
03 August 2025
02
Let Figure M2W2P1 represent the graph of a function
f
f
f
. The solid points denote the value of the function at the points, and the values denoted by the hollow points are not taken by the function. Choose the set of correct options.
Multiple correct
MEDIUM
5 marks
03 August 2025
03
Consider the functions
f
(
x
)
=
{
x
−
6
,
x
<
6
x
−
7
,
x
≥
6
f(x)=\begin{cases}x-6,&x<6\\x-7,&x\ge 6\end{cases}
f
(
x
)
=
{
x
−
6
,
x
−
7
,
x
<
6
x
≥
6
and
g
(
x
)
=
{
(
x
−
6
)
2
,
x
<
6.5
(
x
−
7
)
2
,
x
≥
6.5
g(x)=\begin{cases}(x-6)^2,&x<6.5\\(x-7)^2,&x\ge 6.5\end{cases}
g
(
x
)
=
{
(
x
−
6
)
2
,
(
x
−
7
)
2
,
x
<
6.5
x
≥
6.5
. Which of the following statements is/are correct?
Multiple correct
MEDIUM
4 marks
03 August 2025
04
Find the value of
lim
x
→
∞
(
10
x
−
1
)
(
x
+
3
)
(
x
+
2
)
(
x
+
1
)
x
(
5
x
−
1
)
(
x
+
9
)
(
x
+
4
)
(
x
+
8
)
x
\displaystyle \lim_{x\to\infty}\frac{(10x-1)(x+3)(x+2)(x+1)x}{(5x-1)(x+9)(x+4)(x+8)x}
x
→
∞
lim
(
5
x
−
1
)
(
x
+
9
)
(
x
+
4
)
(
x
+
8
)
x
(
10
x
−
1
)
(
x
+
3
)
(
x
+
2
)
(
x
+
1
)
x
.
Numerical
EASY
2 marks
03 August 2025
05
Let
{
a
n
}
\{a_n\}
{
a
n
}
be a sequence defined as
a
n
=
27
n
4
+
25
n
3
+
2
n
5
+
n
4
n
2
+
2
n
+
10
n
3
+
5
n
5
+
n
a_n=\frac{27n^4+25n^3+2n^5+n}{4n^2+2n+10n^3+5n^5+n}
a
n
=
4
n
2
+
2
n
+
10
n
3
+
5
n
5
+
n
27
n
4
+
25
n
3
+
2
n
5
+
n
. Consider a sequence
{
b
n
}
\{b_n\}
{
b
n
}
defined as
b
n
=
2
5
a
n
b_n=2^{5a_n}
b
n
=
2
5
a
n
. Find the limit of
{
b
n
}
\{b_n\}
{
b
n
}
.
Numerical
MEDIUM
3 marks
03 August 2025
06
The value of
lim
x
→
0
(
1
+
x
)
1
/
2
−
(
1
−
x
)
1
/
2
x
\displaystyle \lim_{x\to0}\frac{(1+x)^{1/2}-(1-x)^{1/2}}{x}
x
→
0
lim
x
(
1
+
x
)
1/2
−
(
1
−
x
)
1/2
is:
Single correct
EASY
4 marks
16 March 2025
07
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
3
n
+
2
−
7
n
3
n
+
6
n
a_n=\frac{3^{n+2}-7n}{3^n+6n}
a
n
=
3
n
+
6
n
3
n
+
2
−
7
n
. Find
lim
n
→
+
∞
a
n
\lim_{n\to+\infty}a_n
lim
n
→
+
∞
a
n
.
Comprehension
MEDIUM
2 marks
16 March 2025
08
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
2
n
+
5
(
−
1
)
n
4
n
−
3
(
−
1
)
n
a_n=\frac{2n+5(-1)^n}{4n-3(-1)^n}
a
n
=
4
n
−
3
(
−
1
)
n
2
n
+
5
(
−
1
)
n
. Find
lim
n
→
+
∞
a
n
\lim_{n\to+\infty}a_n
lim
n
→
+
∞
a
n
.
Comprehension
EASY
2 marks
16 March 2025
09
If the function
f
(
x
)
=
{
A
x
−
B
,
x
≤
−
1
,
2
x
2
+
3
A
x
+
B
,
−
1
<
x
≤
1
,
4
,
x
>
1
f(x)=\begin{cases}Ax-B, & x\le-1,\\ 2x^2+3Ax+B, & -1<x\le1,\\ 4, & x>1\end{cases}
f
(
x
)
=
⎩
⎨
⎧
A
x
−
B
,
2
x
2
+
3
A
x
+
B
,
4
,
x
≤
−
1
,
−
1
<
x
≤
1
,
x
>
1
is continuous for all
x
∈
R
x\in\mathbb{R}
x
∈
R
, then find the value of
6
(
A
+
B
)
6(A+B)
6
(
A
+
B
)
.
Numerical
MEDIUM
4 marks
01 December 2024
10
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
n
3
−
3
n
2
+
sin
n
2
n
3
+
ln
n
+
n
2
a_n=\frac{n^3-3n^2+\sin n}{2n^3+\ln n+n^2}
a
n
=
2
n
3
+
l
n
n
+
n
2
n
3
−
3
n
2
+
s
i
n
n
. Find
lim
n
→
∞
a
n
\lim_{n\to\infty}a_n
lim
n
→
∞
a
n
.
Comprehension
EASY
2 marks
01 December 2024
11
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
e
3
n
+
n
4
e
5
n
+
n
5
a_n=\frac{e^{3n}+n^4}{e^{5n}+n^5}
a
n
=
e
5
n
+
n
5
e
3
n
+
n
4
. Find
lim
n
→
∞
a
n
\lim_{n\to\infty}a_n
lim
n
→
∞
a
n
.
Comprehension
EASY
3 marks
01 December 2024
12
Is the statement True or False: The left-hand limit (LHL) and right-hand limit (RHL) of the given function
f
(
x
)
f(x)
f
(
x
)
exist at
x
=
0
x=0
x
=
0
and are equal to each other.
Comprehension
MEDIUM
2 marks
04 August 2024
13
Is the statement True or False: The function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
0
x=0
x
=
0
.
Comprehension
EASY
2 marks
04 August 2024
14
Find the total number of points in
[
−
3
,
3
]
[-3,3]
[
−
3
,
3
]
at which
f
(
x
)
f(x)
f
(
x
)
is not continuous.
Comprehension
MEDIUM
2 marks
04 August 2024
15
Is the statement True or False: The function
∣
x
+
1
∣
f
(
x
)
|x+1|f(x)
∣
x
+
1∣
f
(
x
)
is continuous at
x
=
−
1
x=-1
x
=
−
1
.
Comprehension
EASY
2 marks
04 August 2024
16
Calculate the limit of the following function at
x
=
0
x=0
x
=
0
:
f
(
x
)
=
x
2
−
2
x
+
4
f(x)=x^2-2x+4
f
(
x
)
=
x
2
−
2
x
+
4
if
x
≥
0
x\ge0
x
≥
0
, and
f
(
x
)
=
e
x
2
+
3
f(x)=e^{x^2}+3
f
(
x
)
=
e
x
2
+
3
if
x
<
0
x<0
x
<
0
.
Comprehension
EASY
2 marks
04 August 2024
17
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
6
+
6
⋅
2
2
+
6
⋅
3
2
+
⋯
+
6
⋅
n
2
4
n
6
+
5
a_n=\frac{6+6\cdot2^2+6\cdot3^2+\cdots+6\cdot n^2}{\sqrt{4n^6+5}}
a
n
=
4
n
6
+
5
6
+
6
⋅
2
2
+
6
⋅
3
2
+
⋯
+
6
⋅
n
2
.
Comprehension
MEDIUM
2 marks
04 August 2024
18
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
100
n
2
−
11
100
n
3
+
7
a_n=\frac{100n^2-11}{100n^3+7}
a
n
=
100
n
3
+
7
100
n
2
−
11
.
Comprehension
EASY
2 marks
04 August 2024
19
Consider the function
f
(
x
)
=
2
x
2
∣
x
∣
f(x)=\frac{2x^2}{|x|}
f
(
x
)
=
∣
x
∣
2
x
2
. Then
lim
x
→
0
f
(
x
)
\lim_{x\to0} f(x)
lim
x
→
0
f
(
x
)
is:
Numerical
EASY
3 marks
24 March 2024
20
Consider the function
f
(
x
)
=
{
3
x
(
x
+
2
)
2
,
x
≤
−
1
,
2
x
−
3
,
−
1
<
x
≤
1
,
−
8
x
+
1
,
x
>
1.
f(x)=\begin{cases}\frac{3x}{(x+2)^2}, & x\le -1,\\ 2x-3, & -1<x\le 1,\\ -\frac{8}{x+1}, & x>1.\end{cases}
f
(
x
)
=
⎩
⎨
⎧
(
x
+
2
)
2
3
x
,
2
x
−
3
,
−
x
+
1
8
,
x
≤
−
1
,
−
1
<
x
≤
1
,
x
>
1.
Find the total number of points in
(
−
2
,
2
]
(-2,2]
(
−
2
,
2
]
at which
f
(
x
)
f(x)
f
(
x
)
is not continuous.
Numerical
MEDIUM
4 marks
24 March 2024
21
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
100
n
2
−
11
100
n
3
+
7
a_n=\frac{100n^2-11}{100n^3+7}
a
n
=
100
n
3
+
7
100
n
2
−
11
. Find
lim
n
→
∞
a
n
\lim_{n\to\infty} a_n
lim
n
→
∞
a
n
.
Comprehension
EASY
4 marks
24 March 2024
22
Evaluate the following limit:
lim
x
→
2
x
6
−
24
x
−
16
x
3
+
2
x
−
12
\lim_{x\to2}\frac{x^6-24x-16}{x^3+2x-12}
lim
x
→
2
x
3
+
2
x
−
12
x
6
−
24
x
−
16
.
Comprehension
MEDIUM
4 marks
24 March 2024
23
Choose the correct option for
f
(
x
)
=
1
x
−
1
f(x)=\frac{1}{x-1}
f
(
x
)
=
x
−
1
1
.
Single correct
MEDIUM
4 marks
03 December 2023
24
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
n
3
+
2
n
2
−
1
n
3
+
3
n
+
1
a_n=\frac{n^3+2n^2-1}{n^3+3n+1}
a
n
=
n
3
+
3
n
+
1
n
3
+
2
n
2
−
1
.
Comprehension
EASY
3 marks
03 December 2023
25
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
n
3
+
2
n
3
n
+
7
n
a_n=\frac{n^3+2^n}{3^n+7^n}
a
n
=
3
n
+
7
n
n
3
+
2
n
.
Comprehension
EASY
3 marks
03 December 2023
26
Given a function
f
(
x
)
=
{
∣
x
∣
x
,
x
≠
0
,
1
,
x
=
0.
f(x)=\begin{cases}\frac{|x|}{x}, & x\ne0,\\ 1, & x=0.\end{cases}
f
(
x
)
=
{
x
∣
x
∣
,
1
,
x
=
0
,
x
=
0.
Which of the following options is/are true?
Multiple correct
MEDIUM
4 marks
03 December 2023
27
Consider a function
f
f
f
defined as
f
(
x
)
=
{
3
m
x
+
n
,
x
<
1
,
11
,
x
=
1
,
5
m
x
−
2
n
,
x
>
1.
f(x)=\begin{cases}3mx+n, & x<1,\\ 11, & x=1,\\ 5mx-2n, & x>1.\end{cases}
f
(
x
)
=
⎩
⎨
⎧
3
m
x
+
n
,
11
,
5
m
x
−
2
n
,
x
<
1
,
x
=
1
,
x
>
1.
If
f
f
f
is continuous at
x
=
1
x=1
x
=
1
, then find the value of
m
+
n
m+n
m
+
n
.
Numerical
MEDIUM
5 marks
03 December 2023
28
Which of the following options is/are true?
Multiple correct
EASY
3 marks
06 August 2023
29
What is the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
?
Comprehension
EASY
2 marks
06 August 2023
30
What is the limit of the sequence
{
b
n
}
\{b_n\}
{
b
n
}
?
Comprehension
EASY
2 marks
06 August 2023
31
Find the value of
b
b
b
.
Comprehension
EASY
2 marks
06 August 2023
32
Define a function
f
(
x
)
=
{
x
−
3
∣
x
−
3
∣
,
x
≠
3
,
1
,
x
=
3.
f(x)=\begin{cases}\frac{x-3}{|x-3|}, & x\ne3,\\ 1, & x=3.\end{cases}
f
(
x
)
=
{
∣
x
−
3∣
x
−
3
,
1
,
x
=
3
,
x
=
3.
Which of the following options is/are true?
Multiple correct
MEDIUM
5 marks
06 August 2023
33
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
2022
+
8
⋅
2023
n
2021
+
4
⋅
2023
n
a_n=\frac{2022+8\cdot2023^n}{2021+4\cdot2023^n}
a
n
=
2021
+
4
⋅
202
3
n
2022
+
8
⋅
202
3
n
.
Comprehension
EASY
2 marks
02 April 2023
34
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
8
n
2
+
10
n
2
n
2
+
6
n
−
7
a_n=\frac{8n^2+10n}{2n^2+6n-7}
a
n
=
2
n
2
+
6
n
−
7
8
n
2
+
10
n
.
Comprehension
EASY
2 marks
02 April 2023
35
Find the limit
lim
x
→
0
4
+
8
x
−
2
x
\lim_{x\to0}\frac{\sqrt{4+8x}-2}{x}
lim
x
→
0
x
4
+
8
x
−
2
.
Comprehension
MEDIUM
2 marks
02 April 2023
36
Find the limit
lim
x
→
0
+
sin
2
x
2
x
\lim_{x\to0^+}\frac{\sin^2 x}{\sqrt{2x}}
lim
x
→
0
+
2
x
s
i
n
2
x
.
Comprehension
MEDIUM
2 marks
02 April 2023
37
Find the number of points where the function
f
(
x
)
f(x)
f
(
x
)
is not continuous.
Comprehension
EASY
2 marks
02 April 2023
38
Find the left limit
lim
x
→
4
−
f
(
x
)
\lim_{x\to4^-}f(x)
lim
x
→
4
−
f
(
x
)
.
Comprehension
EASY
2 marks
02 April 2023
39
Find the right limit
lim
x
→
4
+
10
f
(
x
)
\lim_{x\to4^+}10f(x)
lim
x
→
4
+
10
f
(
x
)
.
Comprehension
EASY
2 marks
02 April 2023
40
Consider a function
f
f
f
defined as
f
(
x
)
=
{
1
e
1
/
x
+
1
,
x
≠
0
0
,
x
=
0.
f(x)=\begin{cases}\frac{1}{e^{1/x}+1},&x\ne0\\0,&x=0.\end{cases}
f
(
x
)
=
{
e
1/
x
+
1
1
,
0
,
x
=
0
x
=
0.
Which of the following option(s) is/are true?
Multiple correct
MEDIUM
5 marks
20 November 2022
41
Consider a sequence
(
1
,
2
,
3
,
4
,
5
,
6
,
…
)
(1,2,3,4,5,6,\ldots)
(
1
,
2
,
3
,
4
,
5
,
6
,
…
)
, that is,
a
n
=
n
a_n=n
a
n
=
n
for all
n
∈
N
n\in\mathbb{N}
n
∈
N
. Which of the following sequences are subsequences of
{
a
n
}
\{a_n\}
{
a
n
}
?
Multiple correct
EASY
2 marks
20 November 2022
42
Consider a function
f
(
x
)
=
{
m
x
2
−
n
,
x
<
1
2
,
x
=
1
x
+
n
,
x
>
1.
f(x)=\begin{cases}mx^2-n,&x<1\\2,&x=1\\x+n,&x>1.\end{cases}
f
(
x
)
=
⎩
⎨
⎧
m
x
2
−
n
,
2
,
x
+
n
,
x
<
1
x
=
1
x
>
1.
If
f
f
f
is continuous at
x
=
1
x=1
x
=
1
, then find the value of
m
+
n
m+n
m
+
n
.
Numerical
EASY
3 marks
20 November 2022
43
Evaluate
lim
x
→
0
sin
3
x
sin
(
x
3
)
\lim_{x\to0}\frac{\sin^3 x}{\sin(x^3)}
lim
x
→
0
s
i
n
(
x
3
)
s
i
n
3
x
.
Comprehension
MEDIUM
3 marks
20 November 2022
44
Evaluate
lim
x
→
0
x
+
tan
x
sin
x
\lim_{x\to0}\frac{x+\tan x}{\sin x}
lim
x
→
0
s
i
n
x
x
+
t
a
n
x
.
Comprehension
EASY
2 marks
20 November 2022
45
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
4
n
−
1
3
n
2
+
9
a_n=\frac{4n-1}{3n^2+9}
a
n
=
3
n
2
+
9
4
n
−
1
.
Comprehension
EASY
2 marks
20 November 2022
46
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
10
n
+
1
+
sin
n
n
a_n=\frac{10n+1+\sin n}{n}
a
n
=
n
10
n
+
1
+
s
i
n
n
if
n
n
n
is odd, and
a
n
=
10
n
−
1
+
cos
n
n
a_n=\frac{10n-1+\cos n}{n}
a
n
=
n
10
n
−
1
+
c
o
s
n
if
n
n
n
is even.
Comprehension
MEDIUM
2 marks
20 November 2022
Showing 46 questions.
Mathematics 1 > Week 7
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Type:
All
Difficulty:
All
Year:
All
46Q
01
The value of
lim
x
→
0
(
1
+
x
)
1
/
2
−
(
1
−
x
)
1
/
2
x
\displaystyle \lim_{x\to 0}\frac{(1+x)^{1/2}-(1-x)^{1/2}}{x}
x
→
0
lim
x
(
1
+
x
)
1/2
−
(
1
−
x
)
1/2
is:
Single correct
EASY
3 marks
03 August 2025
02
Let Figure M2W2P1 represent the graph of a function
f
f
f
. The solid points denote the value of the function at the points, and the values denoted by the hollow points are not taken by the function. Choose the set of correct options.
Multiple correct
MEDIUM
5 marks
03 August 2025
03
Consider the functions
f
(
x
)
=
{
x
−
6
,
x
<
6
x
−
7
,
x
≥
6
f(x)=\begin{cases}x-6,&x<6\\x-7,&x\ge 6\end{cases}
f
(
x
)
=
{
x
−
6
,
x
−
7
,
x
<
6
x
≥
6
and
g
(
x
)
=
{
(
x
−
6
)
2
,
x
<
6.5
(
x
−
7
)
2
,
x
≥
6.5
g(x)=\begin{cases}(x-6)^2,&x<6.5\\(x-7)^2,&x\ge 6.5\end{cases}
g
(
x
)
=
{
(
x
−
6
)
2
,
(
x
−
7
)
2
,
x
<
6.5
x
≥
6.5
. Which of the following statements is/are correct?
Multiple correct
MEDIUM
4 marks
03 August 2025
04
Find the value of
lim
x
→
∞
(
10
x
−
1
)
(
x
+
3
)
(
x
+
2
)
(
x
+
1
)
x
(
5
x
−
1
)
(
x
+
9
)
(
x
+
4
)
(
x
+
8
)
x
\displaystyle \lim_{x\to\infty}\frac{(10x-1)(x+3)(x+2)(x+1)x}{(5x-1)(x+9)(x+4)(x+8)x}
x
→
∞
lim
(
5
x
−
1
)
(
x
+
9
)
(
x
+
4
)
(
x
+
8
)
x
(
10
x
−
1
)
(
x
+
3
)
(
x
+
2
)
(
x
+
1
)
x
.
Numerical
EASY
2 marks
03 August 2025
05
Let
{
a
n
}
\{a_n\}
{
a
n
}
be a sequence defined as
a
n
=
27
n
4
+
25
n
3
+
2
n
5
+
n
4
n
2
+
2
n
+
10
n
3
+
5
n
5
+
n
a_n=\frac{27n^4+25n^3+2n^5+n}{4n^2+2n+10n^3+5n^5+n}
a
n
=
4
n
2
+
2
n
+
10
n
3
+
5
n
5
+
n
27
n
4
+
25
n
3
+
2
n
5
+
n
. Consider a sequence
{
b
n
}
\{b_n\}
{
b
n
}
defined as
b
n
=
2
5
a
n
b_n=2^{5a_n}
b
n
=
2
5
a
n
. Find the limit of
{
b
n
}
\{b_n\}
{
b
n
}
.
Numerical
MEDIUM
3 marks
03 August 2025
06
The value of
lim
x
→
0
(
1
+
x
)
1
/
2
−
(
1
−
x
)
1
/
2
x
\displaystyle \lim_{x\to0}\frac{(1+x)^{1/2}-(1-x)^{1/2}}{x}
x
→
0
lim
x
(
1
+
x
)
1/2
−
(
1
−
x
)
1/2
is:
Single correct
EASY
4 marks
16 March 2025
07
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
3
n
+
2
−
7
n
3
n
+
6
n
a_n=\frac{3^{n+2}-7n}{3^n+6n}
a
n
=
3
n
+
6
n
3
n
+
2
−
7
n
. Find
lim
n
→
+
∞
a
n
\lim_{n\to+\infty}a_n
lim
n
→
+
∞
a
n
.
Comprehension
MEDIUM
2 marks
16 March 2025
08
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
2
n
+
5
(
−
1
)
n
4
n
−
3
(
−
1
)
n
a_n=\frac{2n+5(-1)^n}{4n-3(-1)^n}
a
n
=
4
n
−
3
(
−
1
)
n
2
n
+
5
(
−
1
)
n
. Find
lim
n
→
+
∞
a
n
\lim_{n\to+\infty}a_n
lim
n
→
+
∞
a
n
.
Comprehension
EASY
2 marks
16 March 2025
09
If the function
f
(
x
)
=
{
A
x
−
B
,
x
≤
−
1
,
2
x
2
+
3
A
x
+
B
,
−
1
<
x
≤
1
,
4
,
x
>
1
f(x)=\begin{cases}Ax-B, & x\le-1,\\ 2x^2+3Ax+B, & -1<x\le1,\\ 4, & x>1\end{cases}
f
(
x
)
=
⎩
⎨
⎧
A
x
−
B
,
2
x
2
+
3
A
x
+
B
,
4
,
x
≤
−
1
,
−
1
<
x
≤
1
,
x
>
1
is continuous for all
x
∈
R
x\in\mathbb{R}
x
∈
R
, then find the value of
6
(
A
+
B
)
6(A+B)
6
(
A
+
B
)
.
Numerical
MEDIUM
4 marks
01 December 2024
10
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
n
3
−
3
n
2
+
sin
n
2
n
3
+
ln
n
+
n
2
a_n=\frac{n^3-3n^2+\sin n}{2n^3+\ln n+n^2}
a
n
=
2
n
3
+
l
n
n
+
n
2
n
3
−
3
n
2
+
s
i
n
n
. Find
lim
n
→
∞
a
n
\lim_{n\to\infty}a_n
lim
n
→
∞
a
n
.
Comprehension
EASY
2 marks
01 December 2024
11
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
e
3
n
+
n
4
e
5
n
+
n
5
a_n=\frac{e^{3n}+n^4}{e^{5n}+n^5}
a
n
=
e
5
n
+
n
5
e
3
n
+
n
4
. Find
lim
n
→
∞
a
n
\lim_{n\to\infty}a_n
lim
n
→
∞
a
n
.
Comprehension
EASY
3 marks
01 December 2024
12
Is the statement True or False: The left-hand limit (LHL) and right-hand limit (RHL) of the given function
f
(
x
)
f(x)
f
(
x
)
exist at
x
=
0
x=0
x
=
0
and are equal to each other.
Comprehension
MEDIUM
2 marks
04 August 2024
13
Is the statement True or False: The function
f
(
x
)
f(x)
f
(
x
)
is continuous at
x
=
0
x=0
x
=
0
.
Comprehension
EASY
2 marks
04 August 2024
14
Find the total number of points in
[
−
3
,
3
]
[-3,3]
[
−
3
,
3
]
at which
f
(
x
)
f(x)
f
(
x
)
is not continuous.
Comprehension
MEDIUM
2 marks
04 August 2024
15
Is the statement True or False: The function
∣
x
+
1
∣
f
(
x
)
|x+1|f(x)
∣
x
+
1∣
f
(
x
)
is continuous at
x
=
−
1
x=-1
x
=
−
1
.
Comprehension
EASY
2 marks
04 August 2024
16
Calculate the limit of the following function at
x
=
0
x=0
x
=
0
:
f
(
x
)
=
x
2
−
2
x
+
4
f(x)=x^2-2x+4
f
(
x
)
=
x
2
−
2
x
+
4
if
x
≥
0
x\ge0
x
≥
0
, and
f
(
x
)
=
e
x
2
+
3
f(x)=e^{x^2}+3
f
(
x
)
=
e
x
2
+
3
if
x
<
0
x<0
x
<
0
.
Comprehension
EASY
2 marks
04 August 2024
17
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
6
+
6
⋅
2
2
+
6
⋅
3
2
+
⋯
+
6
⋅
n
2
4
n
6
+
5
a_n=\frac{6+6\cdot2^2+6\cdot3^2+\cdots+6\cdot n^2}{\sqrt{4n^6+5}}
a
n
=
4
n
6
+
5
6
+
6
⋅
2
2
+
6
⋅
3
2
+
⋯
+
6
⋅
n
2
.
Comprehension
MEDIUM
2 marks
04 August 2024
18
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
100
n
2
−
11
100
n
3
+
7
a_n=\frac{100n^2-11}{100n^3+7}
a
n
=
100
n
3
+
7
100
n
2
−
11
.
Comprehension
EASY
2 marks
04 August 2024
19
Consider the function
f
(
x
)
=
2
x
2
∣
x
∣
f(x)=\frac{2x^2}{|x|}
f
(
x
)
=
∣
x
∣
2
x
2
. Then
lim
x
→
0
f
(
x
)
\lim_{x\to0} f(x)
lim
x
→
0
f
(
x
)
is:
Numerical
EASY
3 marks
24 March 2024
20
Consider the function
f
(
x
)
=
{
3
x
(
x
+
2
)
2
,
x
≤
−
1
,
2
x
−
3
,
−
1
<
x
≤
1
,
−
8
x
+
1
,
x
>
1.
f(x)=\begin{cases}\frac{3x}{(x+2)^2}, & x\le -1,\\ 2x-3, & -1<x\le 1,\\ -\frac{8}{x+1}, & x>1.\end{cases}
f
(
x
)
=
⎩
⎨
⎧
(
x
+
2
)
2
3
x
,
2
x
−
3
,
−
x
+
1
8
,
x
≤
−
1
,
−
1
<
x
≤
1
,
x
>
1.
Find the total number of points in
(
−
2
,
2
]
(-2,2]
(
−
2
,
2
]
at which
f
(
x
)
f(x)
f
(
x
)
is not continuous.
Numerical
MEDIUM
4 marks
24 March 2024
21
{
a
n
}
\{a_n\}
{
a
n
}
is such that
a
n
=
100
n
2
−
11
100
n
3
+
7
a_n=\frac{100n^2-11}{100n^3+7}
a
n
=
100
n
3
+
7
100
n
2
−
11
. Find
lim
n
→
∞
a
n
\lim_{n\to\infty} a_n
lim
n
→
∞
a
n
.
Comprehension
EASY
4 marks
24 March 2024
22
Evaluate the following limit:
lim
x
→
2
x
6
−
24
x
−
16
x
3
+
2
x
−
12
\lim_{x\to2}\frac{x^6-24x-16}{x^3+2x-12}
lim
x
→
2
x
3
+
2
x
−
12
x
6
−
24
x
−
16
.
Comprehension
MEDIUM
4 marks
24 March 2024
23
Choose the correct option for
f
(
x
)
=
1
x
−
1
f(x)=\frac{1}{x-1}
f
(
x
)
=
x
−
1
1
.
Single correct
MEDIUM
4 marks
03 December 2023
24
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
n
3
+
2
n
2
−
1
n
3
+
3
n
+
1
a_n=\frac{n^3+2n^2-1}{n^3+3n+1}
a
n
=
n
3
+
3
n
+
1
n
3
+
2
n
2
−
1
.
Comprehension
EASY
3 marks
03 December 2023
25
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
n
3
+
2
n
3
n
+
7
n
a_n=\frac{n^3+2^n}{3^n+7^n}
a
n
=
3
n
+
7
n
n
3
+
2
n
.
Comprehension
EASY
3 marks
03 December 2023
26
Given a function
f
(
x
)
=
{
∣
x
∣
x
,
x
≠
0
,
1
,
x
=
0.
f(x)=\begin{cases}\frac{|x|}{x}, & x\ne0,\\ 1, & x=0.\end{cases}
f
(
x
)
=
{
x
∣
x
∣
,
1
,
x
=
0
,
x
=
0.
Which of the following options is/are true?
Multiple correct
MEDIUM
4 marks
03 December 2023
27
Consider a function
f
f
f
defined as
f
(
x
)
=
{
3
m
x
+
n
,
x
<
1
,
11
,
x
=
1
,
5
m
x
−
2
n
,
x
>
1.
f(x)=\begin{cases}3mx+n, & x<1,\\ 11, & x=1,\\ 5mx-2n, & x>1.\end{cases}
f
(
x
)
=
⎩
⎨
⎧
3
m
x
+
n
,
11
,
5
m
x
−
2
n
,
x
<
1
,
x
=
1
,
x
>
1.
If
f
f
f
is continuous at
x
=
1
x=1
x
=
1
, then find the value of
m
+
n
m+n
m
+
n
.
Numerical
MEDIUM
5 marks
03 December 2023
28
Which of the following options is/are true?
Multiple correct
EASY
3 marks
06 August 2023
29
What is the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
?
Comprehension
EASY
2 marks
06 August 2023
30
What is the limit of the sequence
{
b
n
}
\{b_n\}
{
b
n
}
?
Comprehension
EASY
2 marks
06 August 2023
31
Find the value of
b
b
b
.
Comprehension
EASY
2 marks
06 August 2023
32
Define a function
f
(
x
)
=
{
x
−
3
∣
x
−
3
∣
,
x
≠
3
,
1
,
x
=
3.
f(x)=\begin{cases}\frac{x-3}{|x-3|}, & x\ne3,\\ 1, & x=3.\end{cases}
f
(
x
)
=
{
∣
x
−
3∣
x
−
3
,
1
,
x
=
3
,
x
=
3.
Which of the following options is/are true?
Multiple correct
MEDIUM
5 marks
06 August 2023
33
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
2022
+
8
⋅
2023
n
2021
+
4
⋅
2023
n
a_n=\frac{2022+8\cdot2023^n}{2021+4\cdot2023^n}
a
n
=
2021
+
4
⋅
202
3
n
2022
+
8
⋅
202
3
n
.
Comprehension
EASY
2 marks
02 April 2023
34
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
8
n
2
+
10
n
2
n
2
+
6
n
−
7
a_n=\frac{8n^2+10n}{2n^2+6n-7}
a
n
=
2
n
2
+
6
n
−
7
8
n
2
+
10
n
.
Comprehension
EASY
2 marks
02 April 2023
35
Find the limit
lim
x
→
0
4
+
8
x
−
2
x
\lim_{x\to0}\frac{\sqrt{4+8x}-2}{x}
lim
x
→
0
x
4
+
8
x
−
2
.
Comprehension
MEDIUM
2 marks
02 April 2023
36
Find the limit
lim
x
→
0
+
sin
2
x
2
x
\lim_{x\to0^+}\frac{\sin^2 x}{\sqrt{2x}}
lim
x
→
0
+
2
x
s
i
n
2
x
.
Comprehension
MEDIUM
2 marks
02 April 2023
37
Find the number of points where the function
f
(
x
)
f(x)
f
(
x
)
is not continuous.
Comprehension
EASY
2 marks
02 April 2023
38
Find the left limit
lim
x
→
4
−
f
(
x
)
\lim_{x\to4^-}f(x)
lim
x
→
4
−
f
(
x
)
.
Comprehension
EASY
2 marks
02 April 2023
39
Find the right limit
lim
x
→
4
+
10
f
(
x
)
\lim_{x\to4^+}10f(x)
lim
x
→
4
+
10
f
(
x
)
.
Comprehension
EASY
2 marks
02 April 2023
40
Consider a function
f
f
f
defined as
f
(
x
)
=
{
1
e
1
/
x
+
1
,
x
≠
0
0
,
x
=
0.
f(x)=\begin{cases}\frac{1}{e^{1/x}+1},&x\ne0\\0,&x=0.\end{cases}
f
(
x
)
=
{
e
1/
x
+
1
1
,
0
,
x
=
0
x
=
0.
Which of the following option(s) is/are true?
Multiple correct
MEDIUM
5 marks
20 November 2022
41
Consider a sequence
(
1
,
2
,
3
,
4
,
5
,
6
,
…
)
(1,2,3,4,5,6,\ldots)
(
1
,
2
,
3
,
4
,
5
,
6
,
…
)
, that is,
a
n
=
n
a_n=n
a
n
=
n
for all
n
∈
N
n\in\mathbb{N}
n
∈
N
. Which of the following sequences are subsequences of
{
a
n
}
\{a_n\}
{
a
n
}
?
Multiple correct
EASY
2 marks
20 November 2022
42
Consider a function
f
(
x
)
=
{
m
x
2
−
n
,
x
<
1
2
,
x
=
1
x
+
n
,
x
>
1.
f(x)=\begin{cases}mx^2-n,&x<1\\2,&x=1\\x+n,&x>1.\end{cases}
f
(
x
)
=
⎩
⎨
⎧
m
x
2
−
n
,
2
,
x
+
n
,
x
<
1
x
=
1
x
>
1.
If
f
f
f
is continuous at
x
=
1
x=1
x
=
1
, then find the value of
m
+
n
m+n
m
+
n
.
Numerical
EASY
3 marks
20 November 2022
43
Evaluate
lim
x
→
0
sin
3
x
sin
(
x
3
)
\lim_{x\to0}\frac{\sin^3 x}{\sin(x^3)}
lim
x
→
0
s
i
n
(
x
3
)
s
i
n
3
x
.
Comprehension
MEDIUM
3 marks
20 November 2022
44
Evaluate
lim
x
→
0
x
+
tan
x
sin
x
\lim_{x\to0}\frac{x+\tan x}{\sin x}
lim
x
→
0
s
i
n
x
x
+
t
a
n
x
.
Comprehension
EASY
2 marks
20 November 2022
45
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
4
n
−
1
3
n
2
+
9
a_n=\frac{4n-1}{3n^2+9}
a
n
=
3
n
2
+
9
4
n
−
1
.
Comprehension
EASY
2 marks
20 November 2022
46
Find the limit of the sequence
{
a
n
}
\{a_n\}
{
a
n
}
such that
a
n
=
10
n
+
1
+
sin
n
n
a_n=\frac{10n+1+\sin n}{n}
a
n
=
n
10
n
+
1
+
s
i
n
n
if
n
n
n
is odd, and
a
n
=
10
n
−
1
+
cos
n
n
a_n=\frac{10n-1+\cos n}{n}
a
n
=
n
10
n
−
1
+
c
o
s
n
if
n
n
n
is even.
Comprehension
MEDIUM
2 marks
20 November 2022
Showing 46 questions.
