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Recent questions tagged limits
2
votes
3
answers
61
ISI2014-DCG-17
$\underset{x \to 2}{\lim} \dfrac{1}{1+e^{\frac{1}{x-2}}}$ is $0$ $1/2$ $1$ non-existent
$\underset{x \to 2}{\lim} \dfrac{1}{1+e^{\frac{1}{x-2}}}$ is$0$$1/2$$1$non-existent
Arjun
605
views
Arjun
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
limits
+
–
1
votes
0
answers
62
ISI2014-DCG-33
Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties. $f(0)=0$, $f(1)=1$, and $f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_2<1$. Then the number of such functions is $0$ $1$ $2$ $\infty$
Let $f(x)$ be a continuous function from $[0,1]$ to $[0,1]$ satisfying the following properties.$f(0)=0$,$f(1)=1$, and$f(x_1)<f(x_2)$ for $x_1 < x_2$ with $0 < x_1, \: x_...
Arjun
461
views
Arjun
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
functions
limits
+
–
2
votes
2
answers
63
ISI2014-DCG-37
Let $f: \bigg( – \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg) \to \mathbb{R}$ be a continuous function, $f(x) \to +\infty$ as $x \to \dfrac{\pi^-}{2}$ and $f(x) \to – \infty$ as $x \to -\dfrac{\pi^+}{2}$. Which one of the following functions satisfies the above properties of $f(x)$? $\cos x$ $\tan x$ $\tan^{-1} x$ $\sin x$
Let $f: \bigg( – \dfrac{\pi}{2}, \dfrac{\pi}{2} \bigg) \to \mathbb{R}$ be a continuous function, $f(x) \to +\infty$ as $x \to \dfrac{\pi^-}{2}$ and $f(x) \to – \infty...
Arjun
557
views
Arjun
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
functions
limits
continuity
+
–
0
votes
0
answers
64
ISI2014-DCG-43
Let $f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid -1, & \text{ if } x>0. \end{cases}$ Then $\underset{x \to a}{\lim} f(x)$ exists if $a=0$ for all $a \in R$ for all $a \neq 0$ only if $a=1$
Let $$f(x) = \begin{cases}\mid \:x \mid +1, & \text{ if } x<0 \\ 0, & \text{ if } x=0 \\ \mid \:x \mid -1, & \text{ if } x>0. \end{cases}$$ Then $\underset{x \to a}{\lim}...
Arjun
351
views
Arjun
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
functions
limits
+
–
0
votes
0
answers
65
ISI2014-DCG-50
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1- \cos tx}$ is equal to $0$ $1$ $\infty$ $2$
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1- \cos tx}$ is equal to$0$$1$$\infty$$2$
Arjun
495
views
Arjun
asked
Sep 23, 2019
Calculus
isi2014-dcg
calculus
limits
+
–
3
votes
2
answers
66
ISI2015-MMA-10
The value of the infinite product $P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^3-1}{n^3+1} \times \cdots \text{ is }$ $1$ $2/3$ $7/3$ none of the above
The value of the infinite product$$P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^3-1}{n^3+1} \times \cdots \text{ is }$$$1$$2/3$$7/...
Arjun
889
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
+
–
0
votes
2
answers
67
ISI2015-MMA-19
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k-1) }{n}} \end{vmatrix}\:\:\:$ is $2$ $2e$ $2 \pi$ $2i$
The limit $\:\:\:\underset{n \to \infty}{\lim} \Sigma_{k=1}^n \begin{vmatrix} e^{\frac{2 \pi i k }{n}} – e^{\frac{2 \pi i (k-1) }{n}} \end{vmatrix}\:\:\:$ is$2$$2e$$2 ...
Arjun
904
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
non-gate
+
–
1
votes
3
answers
68
ISI2015-MMA-20
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals $e^{-1}$ $e^{-1/2}$ $e^{-2}$ $1$
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals$e^{-1}$$e^{-1/2}$$e^{-2}$$1$
Arjun
675
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
non-gate
+
–
0
votes
1
answer
69
ISI2015-MMA-22
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1- \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$ equals $1$ does not exist equals $\frac{1}{\sqrt{\pi}}$ equals $0$
Let $a_n= \bigg( 1 – \frac{1}{\sqrt{2}} \bigg) \cdots \bigg( 1- \frac{1}{\sqrt{n+1}} \bigg), \: \: n \geq1$. Then $\underset{n \to \infty}{\lim} a_n$equals $1$does not ...
Arjun
684
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
non-gate
+
–
1
votes
1
answer
70
ISI2015-MMA-25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals$1$$0$$e^{-8/3}$$e^{4/9}$
Arjun
757
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
non-gate
+
–
0
votes
1
answer
71
ISI2015-MMA-26
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$
$\displaystyle{}\underset{n \to \infty}{\lim} \frac{1}{n} \bigg( \frac{n}{n+1} + \frac{n}{n+2} + \cdots + \frac{n}{2n} \bigg)$ is equal to$\infty$$0$$\log_e 2$$1$
Arjun
704
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
non-gate
+
–
0
votes
1
answer
72
ISI2015-MMA-55
Let $\{a_n\}$ be a sequence of real numbers. Then $\underset{n \to \infty}{\lim} a_n$ exists if and only if $\underset{n \to \infty}{\lim} a_{2n}$ and $\underset{n \to \infty}{\lim} a_{2n+2}$ exists $\underset{n \to \infty}{\lim} a_{2n}$ ... $\underset{n \to \infty}{\lim} a_{3n}$ exist none of the above
Let $\{a_n\}$ be a sequence of real numbers. Then $\underset{n \to \infty}{\lim} a_n$ exists if and only if$\underset{n \to \infty}{\lim} a_{2n}$ and $\underset{n \to \in...
Arjun
815
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
+
–
1
votes
1
answer
73
ISI2015-MMA-57
Suppose $a>0$. Consider the sequence $a_n = n \{ \sqrt[n]{ea} – \sqrt[n]{a}, \:\:\:\:\: n \geq 1$. Then $\underset{n \to \infty}{\lim} a_n$ does not exist $\underset{n \to \infty}{\lim} a_n=e$ $\underset{n \to \infty}{\lim} a_n=0$ none of the above
Suppose $a>0$. Consider the sequence $a_n = n \{ \sqrt[n]{ea} – \sqrt[n]{a}, \:\:\:\:\: n \geq 1$. Then$\underset{n \to \infty}{\lim} a_n$ does not exist$\underset{n \t...
Arjun
470
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
+
–
0
votes
0
answers
74
ISI2015-MMA-58
Let $\{a_n\}, n \geq 1$, be a sequence of real numbers satisfying $\mid a_n \mid \leq 1$ for all $n$. Define $A_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$, for $n \geq 1$. Then $\underset{n \to \infty}{\lim} \sqrt{n}(A_{n+1}-A_n)$ is equal to $0$ $-1$ $1$ none of these
Let $\{a_n\}, n \geq 1$, be a sequence of real numbers satisfying $\mid a_n \mid \leq 1$ for all $n$. Define $A_n = \frac{1}{n}(a_1+a_2+\cdots+a_n)$, for $n \geq 1$. Then...
Arjun
446
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
non-gate
+
–
0
votes
1
answer
75
ISI2015-MMA-73
$f(x)$ is a differentiable function on the real line such that $\underset{x \to \infty=}{\lim} f(x) =1$ and $\underset{x \to \infty=}{\lim} f’(x) =\alpha$. Then $\alpha$ must be $0$ $\alpha$ need not be $0$, but $\mid \alpha \mid <1$ $\alpha >1$ $\alpha < -1$
$f(x)$ is a differentiable function on the real line such that $\underset{x \to \infty=}{\lim} f(x) =1$ and $\underset{x \to \infty=}{\lim} f’(x) =\alpha$. Then$\alpha$...
Arjun
467
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
differentiation
+
–
0
votes
2
answers
76
ISI2015-MMA-78
The value of $\displaystyle \lim_{n \to \infty} \left[ (n+1) \int_0^1 x^n \ln(1+x) dx \right]$ is $0$ $\ln 2$ $\ln 3$ $\infty$
The value of $$\displaystyle \lim_{n \to \infty} \left[ (n+1) \int_0^1 x^n \ln(1+x) dx \right]$$ is$0$$\ln 2$$\ln 3$$\infty$
Arjun
493
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
calculus
limits
definite-integral
non-gate
+
–
1
votes
1
answer
77
ISI2015-MMA-81
If $f$ is continuous in $[0,1]$ then $\displaystyle \lim_ {n \to \infty} \sum_{j=0}^{[n/2]} \frac{1}{n} f \left(\frac{j}{n} \right)$ (where $[y]$ is the largest integer less than or equal to $y$) does not exist exists and is equal to $\frac{1}{2} \int_0^1 f(x) dx$ exists and is equal to $ \int_0^1 f(x) dx$ exists and is equal to $\int_0^{1/2} f(x) dx$
If $f$ is continuous in $[0,1]$ then $$\displaystyle \lim_ {n \to \infty} \sum_{j=0}^{[n/2]} \frac{1}{n} f \left(\frac{j}{n} \right)$$ (where $[y]$ is the largest integ...
Arjun
379
views
Arjun
asked
Sep 23, 2019
Calculus
isi2015-mma
limits
definite-integral
non-gate
+
–
1
votes
2
answers
78
ISI2015-DCG-45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan ^2 x – x \tan x }{\sin x}$ is $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $0$ None of these
The value of $\underset{x \to 0}{\lim} \dfrac{\tan ^2 x – x \tan x }{\sin x}$ is$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$0$None of these
gatecse
359
views
gatecse
asked
Sep 18, 2019
Calculus
isi2015-dcg
calculus
limits
+
–
1
votes
1
answer
79
ISI2015-DCG-48
$\underset{x \to 1}{\lim} \dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
$\underset{x \to 1}{\lim} \dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals$\frac{4}{3}$$\frac{3}{4}$$1$None of these
gatecse
357
views
gatecse
asked
Sep 18, 2019
Calculus
isi2015-dcg
calculus
limits
+
–
1
votes
1
answer
80
ISI2015-DCG-52
$\underset{x \to -1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
$\underset{x \to -1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals$\frac{3}{5}$$\frac{5}{3}$$1$$\infty$
gatecse
298
views
gatecse
asked
Sep 18, 2019
Calculus
isi2015-dcg
calculus
limits
+
–
0
votes
1
answer
81
ISI2015-DCG-54
$\underset{x \to 0}{\lim} x \sin \left( \frac{1}{x} \right)$ equals $-1$ $0$ $1$ Does not exist
$\underset{x \to 0}{\lim} x \sin \left( \frac{1}{x} \right)$ equals$-1$$0$$1$Does not exist
gatecse
310
views
gatecse
asked
Sep 18, 2019
Calculus
isi2015-dcg
calculus
limits
+
–
1
votes
0
answers
82
ISI2015-DCG-55
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals $-1$ $0$ $1$ Does not exist
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals$-1$$0$$1$Does not exist
gatecse
404
views
gatecse
asked
Sep 18, 2019
Calculus
isi2015-dcg
calculus
limits
+
–
1
votes
1
answer
83
ISI2015-DCG-56
$\underset{x \to \infty}{\lim} \left( 1 + \dfrac{1}{x^2} \right) ^x$ equals $-1$ $0$ $1$ Does not exist
$\underset{x \to \infty}{\lim} \left( 1 + \dfrac{1}{x^2} \right) ^x$ equals$-1$$0$$1$Does not exist
gatecse
478
views
gatecse
asked
Sep 18, 2019
Calculus
isi2015-dcg
calculus
limits
+
–
0
votes
1
answer
84
ISI2015-DCG-58
$\underset{x \to 1}{\lim} \dfrac{x^{16}-1}{\mid x-1 \mid}$ equals $-1$ $0$ $1$ Does not exist
$\underset{x \to 1}{\lim} \dfrac{x^{16}-1}{\mid x-1 \mid}$ equals$-1$$0$$1$Does not exist
gatecse
327
views
gatecse
asked
Sep 18, 2019
Calculus
isi2015-dcg
calculus
limits
+
–
2
votes
2
answers
85
ISI2016-DCG-45
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:x-x\:\tan\:x}{\sin\:x}$ is $\frac{\sqrt{3}}{2}$ $\frac{1}{2}$ $0$ None of these
The value of $\underset{x \to 0}{\lim} \dfrac{\tan^{2}\:x-x\:\tan\:x}{\sin\:x}$ is$\frac{\sqrt{3}}{2}$$\frac{1}{2}$$0$None of these
gatecse
443
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
limits
+
–
0
votes
1
answer
86
ISI2016-DCG-49
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals $\frac{4}{3}$ $\frac{3}{4}$ $1$ None of these
$\underset{x\rightarrow 1}{\lim}\dfrac{x^{\frac{1}{3}}-1}{x^{\frac{1}{4}}-1}$ equals$\frac{4}{3}$$\frac{3}{4}$$1$None of these
gatecse
244
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
0
votes
1
answer
87
ISI2016-DCG-53
$\underset{x\rightarrow-1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$
$\underset{x\rightarrow-1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals$\frac{3}{5}$$\frac{5}{3}$$1$$\infty$
gatecse
221
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
0
votes
0
answers
88
ISI2016-DCG-54
$\underset{x\rightarrow 0}{\lim}x\sin\left(\dfrac{1}{x}\right)$ equals $-1$ $0$ $1$ Does not exist
$\underset{x\rightarrow 0}{\lim}x\sin\left(\dfrac{1}{x}\right)$ equals$-1$$0$$1$Does not exist
gatecse
226
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
0
votes
0
answers
89
ISI2016-DCG-55
$\underset{x\rightarrow 0}{\lim}\sin\left(\dfrac{1}{x}\right)$ equals $-1$ $0$ $1$ Does not exist
$\underset{x\rightarrow 0}{\lim}\sin\left(\dfrac{1}{x}\right)$ equals$-1$$0$$1$Does not exist
gatecse
239
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
0
votes
1
answer
90
ISI2016-DCG-56
$\underset{x\rightarrow \infty}{\lim} \left(1+\dfrac{1}{x^{2}}\right)^{x}$ equals $-1$ $0$ $1$ Does not exist
$\underset{x\rightarrow \infty}{\lim} \left(1+\dfrac{1}{x^{2}}\right)^{x}$ equals$-1$$0$$1$Does not exist
gatecse
313
views
gatecse
asked
Sep 18, 2019
Calculus
isi2016-dcg
calculus
limits
+
–
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