Register

Recent questions tagged limits

1 1 vote
1 1 answer
459
459 views
TIFR-2016-Maths-B: 23
Let $\left \{ f_{n} \right \}^{\infty }_{n=1}$ be the sequence of functions on $\mathbb{R}$ defined by $f_{n}\left ( x \right )=n^{2}x^{n}.$. Let $A$ be the set of all po...
User Avatar
0 0 votes
0 0 answers
445
445 views
TIFR-2016-Maths-A: 5
Which of the following continuous functions $f:\left ( 0,\infty \right ) \rightarrow \mathbb{R}$ can be extended to a continuous function on $\left [ 0,\infty \right )$ ?...
User Avatar
0 0 votes
0 0 answers
348
348 views
TIFR-2016-Maths-A: 15
Which of the following sequences of functions $\left \{ f_{n} \right \}_{n=1}^{\infty }$ converges uniformly ?$f_{n}\left ( x \right )=x^{n}\:on \: \left [ 0,1 \right ]$$...
User Avatar
0 0 votes
0 0 answers
348
348 views
TIFR-2016-Maths-A: 19
The limit$$\underset{n\rightarrow \infty }{lim}\left ( \frac{1}{n} +\frac{1}{n+1}+\dots +\frac{1}{2n}\right )=$$$e$$2$$log_{e}2$$e^{2}$
User Avatar
0 0 votes
0 0 answers
330
330 views
TIFR-2017-Maths-A: 4
True/Flase Question :Over the real line,$$\underset{x\rightarrow \infty }{lim}\: log\left ( 1+\sqrt{4+x} -\sqrt{1+x}\right )=log\left ( 2 \right ).$$
User Avatar
0 0 votes
0 0 answers
348
348 views
TIFR-2017-Maths-A: 5
True/False Question :Suppose $f$ is a continuously differentiable function on $\mathbb{R}$ such that $f\left ( x \right )\rightarrow 1$ and ${f}'\left ( x \right )\right...
User Avatar
0 0 votes
0 0 answers
311
311 views
TIFR-2017-Maths-A: 10
True/False Question :There exists a non-negative continuous function $f:\left [ 0,1 \right ]\rightarrow \mathbb{R}$ such that $\int_{0}^{1}f^{n}dx\rightarrow 2$ as $n\rig...
User Avatar
0 0 votes
0 0 answers
262
262 views
TIFR-2017-Maths-A: 17
True/False Question :Over the real line, $$\underset{x\rightarrow \infty }{lim}\left ( \frac{x+log\:9}{x-log\:9} \right )^{x}=81.$$
User Avatar
0 0 votes
0 0 answers
266
266 views
TIFR-2017-Maths-A: 25
True/False Question :For any $x \in \mathbb{R}$, the sequence $\left \{ a_{n} \right \}$, where $a_{1}=x$ and $a_{n+1}=cos\left ( a_{n} \right )$ for all $n$, is converge...
User Avatar
0 0 votes
0 0 answers
250
250 views
TIFR-2017-Maths-A: 28
True/False Question :If a particle moving on the Euclidean line traverses distance $1$ in time $1$ starting and ending at rest, then at some time $t \in \left [ 0,1 \righ...
User Avatar
0 0 votes
1 1 answer
608
608 views
TIFR-2017-Maths-A: 29
True/False Question :Let $y\left ( t \right )$ be a real valued function defined on the real line such that ${y}'=y \left ( 1-y \right )$, with $y\left ( 0\right ) \in \l...
User Avatar
0 0 votes
0 0 answers
340
340 views
TIFR-2018-Maths-B: 13
For a sequence $\left \{ a_{n} \right \}$ of real numbers, which of the following is a negation of the statement ‘$\underset{n\rightarrow \infty }{lim}\:a_{n}=0$’?There e...
User Avatar
0 0 votes
0 0 answers
359
359 views
TIFR-2018-Maths-B: 14
Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be continuous. Then which of the following statements implies that $f\left ( 0 \right )=0$?$\underset{n\rightarrow \infty }{lim}\...
User Avatar
0 0 votes
0 0 answers
370
370 views
TIFR-2018-Maths-A: 2
True/False Question :$\underset{x\rightarrow 0}{lim}\:\frac{sin\:x}{log\left ( 1+tan\:x \right )}=1$.
User Avatar
0 0 votes
0 0 answers
365
365 views
TIFR-2018-Maths-A: 4
True/False Question :Let $f$ be a nonnegative continuous function on $\mathbb{R}$ such that $\int_{0}^{\infty }f\left ( t \right )dt$ is finite. Then $\underset{x\rightar...
User Avatar
0 0 votes
0 0 answers
322
322 views
TIFR-2018-Maths-A: 8
True/False Question ;Let $f$ be a twice differentiable function on $\mathbb{R}$ such that both $f$ and ${f}''$ are strictly positive on $\mathbb{R}$. Then $\underset{x\r...
User Avatar
0 0 votes
0 0 answers
294
294 views
TIFR-2018-Maths-A: 10
True/False Question :Let $g$ be a continuous function on $\left [ 0,1 \right ]$ such that $g\left ( 1 \right )=0$. Then the sequence of functions $f_{n}\left ( x \right )...
User Avatar
0 0 votes
0 0 answers
341
341 views
TIFR-2019-Maths-B: 11
True/False Question :If $0$ is a limit point of a set $A\subseteq \left ( 0,\infty \right )$, then the set of all $x\in\left ( 0,\infty \right )$ that can be expressed a...
User Avatar
0 0 votes
0 0 answers
348
348 views
TIFR-2019-Maths-A: 2
For $n\geq 1$, the sequence $\left \{ x_{n} \right \}^{\infty }_{n=1},$ where:$$x_{n}=1+\frac{1}{\sqrt{2}}+\dots+\frac{1}{\sqrt{n}}-2\sqrt{n}$$isdecreasingincreasingconst...
User Avatar
0 0 votes
0 0 answers
364
364 views
TIFR-2019-Maths-A: 3
Define a function:$$f\left ( x \right )=\left\{\begin{matrix} x +x^{2} cos\left ( \frac{\pi}{x} \right ), & x\neq 0\\ 0,& x=0. \end{matrix}\right.$$Consider the following...
User Avatar
0 0 votes
0 0 answers
948
948 views
TIFR-2019-Maths-A: 6
The limit$$\underset{n\rightarrow \infty }{\lim}\:n^{2}\int_{0}^{1}\:\frac{1}{\left ( 1+x^{2} \right )^{n}}\:dx$$is equal to$1$$0$$+\infty$$1/2$
User Avatar
0 0 votes
0 0 answers
302
302 views
TIFR-2019-Maths-A: 11
Let $f$ be a continuous function on $\left [ 0,1 \right ]$. Then the limit $\underset{n\rightarrow \infty }{lim}\int ^{1}_{0}nx^{n} f\left ( x \right )dx$ is equal to $f(...
User Avatar
0 0 votes
0 0 answers
269
269 views
TIFR-2019-Maths-A: 13
Let the sequence $\left \{ x_{n} \right \}_{n\rightarrow 1}^{\infty }$ be defined by $x1=\sqrt{2}$ and $x_{n+1}=\left ( \sqrt{2} \right )^{x_{n}}$ for $n\geq 1$. Then wh...
User Avatar
Page: