Questions on the evaluation of limits.

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1answer
26 views

Choice of numbers in calculating $\lim_{x\rightarrow 0}\frac{e^{2x}-1}{\sin 3x}$

I'm studying the limits to $\lim_{x\rightarrow 0}\frac{e^{2x}-1}{\sin 3x}$. According to my text book, I should calculate it like this: $$\frac{e^{2x}-1}{\sin 3x} = \frac{e^{2x}-1}{2x} \times ...
0
votes
2answers
25 views

Please find the range of $V$ when $V = \frac {1} {X}$ & $0 <x<1 $

if $0 <x<1 $ then what is the range of $V$ when $V = \frac {1} {X}$ i tried to compute it by : when $x=0$ then $V = \frac {1} {X}= \frac {1} {0} =undefined$ when $x=1$ then $V = \frac {1} ...
2
votes
1answer
27 views

Range of the distribution of $(1-X)$ when $X$ follows Beta distribution as $X\sim beta(p,q)$

if $X$ follows beta distribution with parameter $p$ and $q$ where $p>0\quad , q>0$ then $1-X$ follows beta distribution with parameters $q$ and $p$, that is if $X\sim beta(p,q)$ then ...
2
votes
0answers
22 views

Finding limits with substitution

$\lim_{x\to0+}(\sinh(x))^{1/x}$ I started by setting $y=\frac{\sinh(x)}{x}$ and taking the natural logarithm of both sides and trying to solve the limit for $ln(y)$ but I got stuck trying to solve ...
3
votes
1answer
42 views

Prove that $\sin^2{x}+\sin{x^2}$ isn't periodic by using uniform continuity

Before the problem is a proof that says a periodic function whose domain is $\mathbb{R}$ is uniformly continuous.So actually the problem is to prove $\sin^2{x}+\sin{x^2}$ isn't uniformly continuous.I ...
2
votes
2answers
49 views

What can I do this cos term to remove the divide by 0?

I was asked to help someone with this problem, and I don't really know the answer why. But I thought I'd still try. $$\lim_{t \to 10} \frac{t^2 - 100}{t+1} \cos\left( \frac{1}{10-t} \right)+ 100$$ ...
3
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4answers
56 views

calculate $\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}$

How to calculate $$\lim_{t\rightarrow1^+}\frac{\sin(\pi t)}{\sqrt{1+\cos(\pi t)}}$$? I've tried to use L'Hospital, but then I'll get $$\lim_{t\rightarrow1^+}\frac{\pi\cos(\pi t)}{\frac{-\pi\sin(\pi ...
1
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2answers
54 views

How to calculate $\lim\limits_{x\to1^+}\frac{1}{(x-1)^2} \int\limits_{1}^{x} \sqrt{1+\cos(\pi t)}\,\mathrm dt$

Can anyone help me by calculating this limit? I know that I need L'Hôpital but how? $$ \lim_{x \to 1^+}\frac{1}{(x-1)^2} \int_{1}^{x} \sqrt{1+\cos(\pi t)} \,\mathrm dt $$ Thank you very much!!
1
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1answer
28 views

Differentiation - Limits Equal (Possibly MVT, Rolle's or L'Hopital)

Got a quick question from a past exam paper. If $f:R \rightarrow R$ is differentiable, and $f$ is such that $\lim_{ x\rightarrow \infty } f(x)=\lim_{ x \rightarrow -\infty} f(x)=0$ and there is a ...
1
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2answers
47 views

$\lim_{x\to 0} \frac{\sin(t\sqrt{x^2-k^2})}{\sqrt{x^2-k^2}}=?$

$\lim_{x\to 0} \frac{\sin(t\sqrt{x^2-k^2})}{\sqrt{x^2-k^2}}=?$ Anyone can help, please. Does is equal to $t$?
-1
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0answers
61 views

What is $\lim\limits_{|a| \to \infty} \int_0^1 (G(x) - a_0 - a_1x)^2\,dx$?

Assume the integral of g from 0 to 1 is a finite #. $$\lim_{|a| \to \infty} f(a) = \lim_{|a| \to \infty}\int_0^1 (G(x) - a_0 - a_1x)^2\,dx$$ $a= [a_0, a_1]$, as $|a| \to \infty$, we have $a_0^2 + ...
0
votes
1answer
52 views

Probability of two random n-digit numbers dividing each other

Let $n$ be a positive integer. Suppose $a$ and $b$ are randomly (and independently) chosen two $n$-digit positive integers which consist of digits 1, 2, 3, ..., 9. (So in particular neither $a$ nor ...
1
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2answers
47 views

Convergence of these series

$$\sum_{n=1}^\infty \frac{2nx^{n}}{(n+1)^{2}3^{n}} \tag{1}$$ $$\sum_{n=1}^\infty x^{n}\tan\frac{x}{4^{n}}\tag{2}$$ Is there any good article that describes an equivalents like if $$ ...
2
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4answers
63 views

The value of $\lim_{x\to +\infty} \dfrac{e^{x^2}}{10^{|x|}}$

I want to find the value of $$\lim_{x\to +\infty} \dfrac{e^{x^2}}{10^{|x|}}.$$ Since $x \rightarrow + \infty$, I only consider the value of the function for $x \ge 0$, i.e. $$\lim_{x\to +\infty} ...
0
votes
1answer
56 views

Compute limit of the sequence given by $x_1=1$, $x_{n+1}^2=\frac{x_n+3}{2}$

Let $\left(x_n\right)$ be a real sequence such that $x_1=1,x_{n+1}^2=\dfrac{x_n+3}{2},\forall n\geq 1$.Compute $\lim_{n\to+\infty} 3^n\sqrt{\dfrac{9}{4}-x_n^2}$
2
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1answer
25 views

For the following monic polynomial,$f$ of even degree how to prove that that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$

For any monic polynomial $f \in \mathbb {Q[x]}$ of even degree,how to prove, there exists polynomial $g \in \mathbb {Q[x]}$ such that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
0
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3answers
37 views

Evaluating $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$

I am trying to evaluate $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$. Factoring and using partial fraction decomposition, I have found that the indefinite integral is: $$\ln{|x-5|}-\ln{|x-3|} + C$$ But ...
5
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4answers
109 views

Evaluating $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$

Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$? I've calculated that the recurrence relation for this integral is: $\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} ...
4
votes
3answers
92 views

Proving if $f(x)$ is differentiable at $x = x_0$ then $f(x)$ is continuous at $x = x_0$.

Please, see if I made some mistake in the proof below. I mention some theorems in the proof: The condition to $f(x)$ be continuous at $x=x_0$ is $\lim\limits_{x\to x_0} f(x)=f(x_0)$. (1) If $f(x)$ ...
5
votes
5answers
114 views

Limit as $x$ approaches $1$ from the right of $\frac{1}{\ln x}-\frac{1}{x-1}$

$$ \lim_{x\rightarrow 1^+}\;\frac{1}{\ln x}-\frac{1}{x-1} $$ So I would just like to know how to begin to solve this limit, or what topic does this problem fall under so that I can search for ...
6
votes
2answers
134 views

High school contest question

Some work on it reveals the possibility of using gamma function. Is there any easy way to compute it? $$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$
2
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1answer
32 views

Question about limits with variable on exponent

So I have to find the following limit $$\lim_{n\to\infty}\left(1+\frac{2}{n}\right)^{1/n}.$$I said that this is ...
10
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6answers
119 views

A limit on binomial coefficients

Let $$x_n=\frac{1}{n^2}\sum_{k=0}^n \ln\left(n\atop k\right).$$ Find the limit of $x_n$. What I can do is just use Stolz formula. But I could not proceed.
1
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1answer
25 views

$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$

For $f\in L^{p}$, $p \in [1,\infty)$ we want to prove: $$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$ I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
4
votes
3answers
104 views

Limit. $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$

I want to compute $$\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$$ Is it OK how I want to do? ...
3
votes
2answers
69 views

Limit: $\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$

How can I find the following limit? $$\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$$ It's a limit of type $\displaystyle 1^{\infty}$ and if I note with $\displaystyle ...
2
votes
2answers
56 views

Proof f(x) is continuous given $x$ rational and irrational.

How can I resolve the task below: Given $f(x)= \begin{cases} x, &x\in \mathbb{Q}\text{ }\\ 1-x, &x\notin \mathbb{Q}\text{ (irrational)} \end{cases}$, $0 \leq x \leq 1$. Show $f(x)$ is ...
4
votes
8answers
137 views

Limit of $\lim_{x \to 0}\left (x\cdot \sin\left(\dfrac{1}{x}\right)\right)$ is $0$ or $1$?

WolframAlpha says $\lim_{x \to 0} x\cdot \sin\left(\dfrac{1}{x}\right)=0$ but I've found it $1$ as below: $$ \lim_{x \to 0} \left(x\cdot \sin\left(\dfrac{1}{x}\right)\right) = \lim_{x \to 0} ...
2
votes
4answers
85 views

How to evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?

How can I evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?
0
votes
1answer
62 views

Why do these trig functions “overpower” each other?

For example, $\sin(x)\cos(x)$ can be written as $\sin(2x)/2$, the limit as $x$ approaches $0$ of $\sin(x)\cos(x)$ is $0$, and the limit as x approaches $\pi/2$ is $0$. I don't see a reason why sine ...
0
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1answer
45 views

limit of an exponential function

I was trying to understand how we can approximate exp. One example is: $$ \exp(t) = \sum_{i=0}^\infty t^i/i! $$ however, why is the next true: $$\lim_{x\to \infty}\exp \left ({\frac{t^2}{2!} ...
2
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3answers
86 views

What does this mean: there exist an integer N such that $n\ge N$?

I'm reading Rudin's, Principles of Mathematical Analysis, and I keep tripping over this phrase. Usually the phrase by that it implies a some equation with n being the index, subscript, of a point. My ...
1
vote
1answer
19 views

Decreasing from the horizontal asymptote

The function $f(x) = x^2/(x^2 - x -2)$ has the following graph. It has a horizontal asymptote $y=1$. For $x$ less than $-4$, the function is decreasing and its graph is under the asymptote. How is ...
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3answers
30 views

Finding the limit of a particular function

Can not understand the following limit in a past paper. $$\dfrac{1}{[n(e^{\frac{1}{n}}-1)]}\rightarrow 1$$
3
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7answers
125 views

How to prove that $1/n!$ is less than $1/n^2$?

I want to prove $$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test. How to prove ...
1
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2answers
50 views

How to find the limit for the quotient of the least number $K_n$ such that the partial sum of the harmonic series $\geq n$

Let $$S_n=1+1/2+\cdots+1/n.$$ Denote by $K_n$ the least subscript $k$ such that $S_k\geq n$. Find the limit $$\lim_{n\to\infty}\frac{K_{n+1}}{K_n}\quad ?$$
1
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3answers
47 views

Evaluating a limit with variable in the exponent

For $$\lim_{x \to \infty} \left(1- \frac{2}{x}\right)^{\dfrac{x}{2}}$$ I have to use the L'Hospital"s rule, right? So I get: $$\lim_{x \to \infty}\frac{x}{2} \log\left(1- \frac{2}{x}\right)$$ And ...
2
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1answer
49 views

How to place a limit that it's inside the integral, outside.

I did this: $$\int_{1}^t x^{-1}dx=\int_{1}^t\lim_{n\rightarrow -1}{x^n}dx =\lim_{n\rightarrow -1}\int_{1}^t{x^n}dx $$ just to have a way to approximate $\ln t$. $$\ln{t}=\lim_{h\rightarrow ...
3
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0answers
59 views

Find the limit $\lim_{n\to\infty}\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}}$ [duplicate]

Find the limit $$\lim_{n\to\infty}\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}}$$. Remark:there are n times square root within $n$.
2
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5answers
44 views

How to find the limit of this function

We have the function $$\dfrac{\sqrt{n^4 + 100}}{4n}$$ I think the best method is by dividing by $n$, but I have no idea what that yields, mainly because of the square root.
5
votes
4answers
121 views

What is $\lim_{n \rightarrow \infty}\sum\limits_{k = 1}^n\frac{k}{n^2}$?

We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$ What is the limit of this function as $n \rightarrow \infty$? My idea: $$\dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + ...
1
vote
1answer
29 views

Find the limit as n approaches infinite

We have the following function: $$U_n = \sin \dfrac{1}{3} n \pi$$ What is the limit of this function as n approaches infinity? I first tried to use my calculator as help, for n I chose some ...
0
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3answers
34 views

Sequence of the ratio of two successive terms of a sequence

If $(a_n)_{n\in N}$ is a strictly decreasing sequence of real number converging to $0$ and s.t. $\forall n\in N$, $0<a_n<1$, does the following limit: $$ \lim_{n}\frac{a_{n+1}}{a_n} $$ exists? ...
2
votes
2answers
45 views

How to calculate $\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}}$ when $x>1$?

Numerically, it looks that the limit is $$\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}} = \frac{1}{\sqrt{2}}\log(1 - x) + cte $$, but I have not been able to ...
4
votes
2answers
32 views

limit of convex increasing funtion

If $f$ is strictly increasing and strictly convex (or $f'>0$ & $f''>0$), then $$\lim_{x\rightarrow∞}{f(x)}=∞$$ Is this statement true? If this statement is true, how can I prove?
8
votes
2answers
93 views

Which is the better approximation to $e$?

Let $a_n = (1+1/n)^n$ and $b_n = (1+1/n)^{n+1}$. Both $a_n \to e$ and $b_n \to e$, and $a_n < e < b_n$. A better approximation to $e$ is known to be $c_n = (1+1/n)^{n+1/2} = \sqrt{a_n b_n} $, ...
3
votes
2answers
44 views

multivariable limit question

Is this an acceptable solution? $$\lim_{(x,y)\rightarrow(0,0)}\frac{\sin(2(x^2+y^2))}{x^2+y^2}$$ $$t=x^2+y^2$$ So $t\rightarrow0$. Now I change the limit to: ...
1
vote
3answers
52 views

Limit as N goes to Infinity

Consider this limit: $$\lim_{n\rightarrow\infty} \left( 1+\frac{1}{n} \right) ^{n^2} = x$$ I thought the way to solve this for $x$ was to reduce it using the fact that as $n \rightarrow \infty$, ...
2
votes
2answers
110 views

Without calculating limit directly show that it is equal to zero

$$\lim_{n\rightarrow\infty}\left(\frac{n+1}{n}\right)^{n^2}\frac{1}{3^n}=0$$ I am not really sure what it means by "without calculating limit" and I don't really have ideas how to do it.
2
votes
3answers
107 views

How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$?

How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$ ? i know that its "1", but why? How can i calculate this? Thank you very very much =)

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