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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

Arjun asked in Calculus Sep 23, 2019

by Arjun

614 views

1 vote

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 $0$ $1$ $2$ $\infty$

Arjun asked in Calculus Sep 23, 2019

by Arjun

466 views

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 $f(x)$? $\cos x$ $\tan x$ $\tan^{-1} x$ $\sin x$

Arjun asked in Calculus Sep 23, 2019

by Arjun

560 views

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 $a=1$

Arjun asked in Calculus Sep 23, 2019

by Arjun

353 views

0 votes

0 answers

65

ISI2014-DCG-50
$\underset{x \to 0}{\lim} \dfrac{x \tan x}{1- \cos tx}$ is equal to $0$ $1$ $\infty$ $2$

Arjun asked in Calculus Sep 23, 2019

by Arjun

501 views

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

Arjun asked in Calculus Sep 23, 2019

by Arjun

893 views

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 $2$ $2e$ $2 \pi$ $2i$

Arjun asked in Calculus Sep 23, 2019

by Arjun

928 views

1 vote

3 answers

68

ISI2015-MMA-20
The limit $\underset{n \to \infty}{\lim} \left( 1- \frac{1}{n^2} \right) ^n$ equals $e^{-1}$ $e^{-1/2}$ $e^{-2}$ $1$

Arjun asked in Calculus Sep 23, 2019

by Arjun

680 views

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 $0$

Arjun asked in Calculus Sep 23, 2019

by Arjun

687 views

1 vote

1 answer

70

ISI2015-MMA-25
The limit $\displaystyle{}\underset{x \to \infty}{\lim} \left( \frac{3x-1}{3x+1} \right) ^{4x}$ equals $1$ $0$ $e^{-8/3}$ $e^{4/9}$

Arjun asked in Calculus Sep 23, 2019

by Arjun

769 views

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 $\infty$ $0$ $\log_e 2$ $1$

Arjun asked in Calculus Sep 23, 2019

by Arjun

711 views

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

Arjun asked in Calculus Sep 23, 2019

by Arjun

824 views

1 vote

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

Arjun asked in Calculus Sep 23, 2019

by Arjun

475 views

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

Arjun asked in Calculus Sep 23, 2019

by Arjun

450 views

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 $\mid \alpha \mid <1$ $\alpha >1$ $\alpha < -1$

Arjun asked in Calculus Sep 23, 2019

by Arjun

471 views

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 $0$ $\ln 2$ $\ln 3$ $\infty$

Arjun asked in Calculus Sep 23, 2019

by Arjun

499 views

1 vote

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 $\int_0^{1/2} f(x) dx$

Arjun asked in Calculus Sep 23, 2019

by Arjun

380 views

1 vote

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

gatecse asked in Calculus Sep 18, 2019

367 views

1 vote

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

gatecse asked in Calculus Sep 18, 2019

363 views

1 vote

1 answer

80

ISI2015-DCG-52
$\underset{x \to -1}{\lim} \dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$

gatecse asked in Calculus Sep 18, 2019

301 views

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

gatecse asked in Calculus Sep 18, 2019

320 views

1 vote

0 answers

82

ISI2015-DCG-55
$\underset{x \to 0}{\lim} \sin \bigg( \dfrac{1}{x} \bigg)$ equals $-1$ $0$ $1$ Does not exist

gatecse asked in Calculus Sep 18, 2019

422 views

1 vote

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

gatecse asked in Calculus Sep 18, 2019

486 views

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

gatecse asked in Calculus Sep 18, 2019

342 views

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

gatecse asked in Calculus Sep 18, 2019

467 views

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

gatecse asked in Calculus Sep 18, 2019

249 views

0 votes

1 answer

87

ISI2016-DCG-53
$\underset{x\rightarrow-1}{\lim}\dfrac{1+\sqrt[3]{x}}{1+\sqrt[5]{x}}$ equals $\frac{3}{5}$ $\frac{5}{3}$ $1$ $\infty$

gatecse asked in Calculus Sep 18, 2019

224 views

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

gatecse asked in Calculus Sep 18, 2019

232 views

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

gatecse asked in Calculus Sep 18, 2019

246 views

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

gatecse asked in Calculus Sep 18, 2019

320 views

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Recent questions tagged limits