Questions on the evaluation of limits.
4
votes
2answers
33 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 $\lim_{x\to x_0} f(x)=f(x_0)$.
If $f(x)$ is ...
5
votes
5answers
96 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
109 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
votes
1answer
30 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 ...
9
votes
5answers
88 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
vote
1answer
19 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
62 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
60 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
53 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
130 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:
$$
\lim_{x \to 0} \left(x\cdot \sin\left(\dfrac{1}{x}\right)\right) = \lim_{x \to 0} ...
2
votes
4answers
84 views
How to evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?
How can I evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?
0
votes
1answer
59 views
Why do these trig functions “overpower” each other?
For example, $\sin(x)\cos(x)$ can be written as $\sin(2x)/2$, the limit as $x$ approaches $0$ of $\sin(x)\cos(x)$ is $0$, and the limit as x approaches $\pi/2$ is $0$. I don't see a reason why sine ...
0
votes
1answer
45 views
limit of an exponential function
I was trying to understand how we can approximate exp.
One example is:
$$ \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
votes
3answers
84 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
17 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 ...
1
vote
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
votes
7answers
124 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
vote
2answers
49 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
vote
3answers
46 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
votes
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 $\ln t$. $$\ln{t}=\lim_{h\rightarrow ...
3
votes
0answers
58 views
Find the limit $\lim_{n\to\infty}\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}}$ [duplicate]
Find the limit $$\lim_{n\to\infty}\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}}$$. Remark:there are n times square root within $n$.
2
votes
5answers
42 views
How to find the limit of this function
We have the function $$\dfrac{\sqrt{n^4 + 100}}{4n}$$
I think the best method is by dividing by $n$, but I have no idea what that yields, mainly because of the square root.
5
votes
4answers
117 views
What is $\lim_{n \rightarrow \infty}\sum\limits_{k = 1}^n\frac{k}{n^2}$?
We have $$\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
votes
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
42 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
31 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
92 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
43 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
106 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 =)
7
votes
1answer
94 views
What is this limit called? Is it a different kind of derivative?
(first I should notice you this is not something I can look up in a textbook, because I'm learning partial derivatives, alike I do with most Maths, as a hobby. If something below is wrong, blame the ...
0
votes
1answer
14 views
Ignoring exponential terms in asymptotic matching of two point boundary value ODE
So I'm not sure how much background I need to give to set up this question. But in my lecture notes I have that
$e^{-\eta / \epsilon^{1-\alpha}}$
can be ignored where $\epsilon << 1$ and $0 ...
0
votes
2answers
42 views
Find the limit of $2+\left(-\frac{2}{e}\right)^n$, as $n\to\infty$, if it exsists
I'm absolutely unsure about how to approach this. I've considered changing it to $-2=\left(-\frac{2}{e}\right)^n$ and then using the properties of lograrithms, but $\ln(-2)$ is undefined, as is ...
1
vote
2answers
36 views
Is it ever proper to say that the limit of a function equals infinity?
If I calculate a limit and get the value $\infty$, what is the proper way to communicate this? Can I say that the $\lim_{n\to\infty}a_n=\infty$ and therefore the sequence $\{a_n\}$ diverges, or do I ...
3
votes
3answers
27 views
Limit as n approaches infinity involving roots
$$\lim_{n\to\infty}\frac{n}{1+2\sqrt{n}}$$
Given my understanding of how to solve these problems, I need to take the highest power of $n$ in the denominator and then divide both the numerator and ...
3
votes
1answer
51 views
accumulation point of recursive sequence
Given is a sequence with:
$(a_0)=1$, $(a_1=1)$, $a_{n+2}=\frac{1+a_{n+1}}{a_n}$
I now have to show what the accumulation points are:
I guess that the sequence is jumping from number to number like ...
18
votes
4answers
568 views
Find $\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$
Find$$\lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}.$$
This is a recent exam question, which I couldn't figure out in the exam. My guess is it doesn't ...
3
votes
1answer
39 views
Show that sequence approaches fixed point of a function
Problem
Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, $x_{n+1} = ...
4
votes
1answer
138 views
What is the most elementary proof that $\lim_{n \to \infty} (1+1/n)^n$ exists?
Here is my candidate for
the most elementary proof that $\lim_{n \to \infty}(1+1/n)^n $ exists.
I would be interested in seeing others.
$***$ Added after some comments:
I prove here by very ...
0
votes
1answer
38 views
please,find the limit of $x$
please,find the limit of $x$ when
$x=u^{-\frac{1}{\alpha}}$
where ${0}\le u\le{1}$ and $\alpha>0$
i have found the limit of $x$ is also ${0}\le x\le{1}$
when $u=0$ then $x=0$ since ...
3
votes
1answer
46 views
How to calculate $\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$?
How can we compute the following limit:
$$\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$$
Mathematica gives the answer $\sqrt{e}$. However, I do not know to do it.
2
votes
1answer
61 views
Solving the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$
Suppose the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$ as $f(t)=\frac{e^t}{t}$ and $n\in \mathbb{N} \setminus{0}$.
How to prove that:
The equation above has a unique solution $U_n$ ...
1
vote
2answers
18 views
Limit with variable: non-defined expression
I have a given limit that depends on a variable $a$:
$$\lim_{x \rightarrow \infty} \left (\frac{e^{ax}}{1 - ax} \right)$$
I understand cases for $a < 0 \implies \lim = 0$ and $a > 0 \implies ...
7
votes
2answers
88 views
Finding $\lim\limits_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$ for continuous $f:[0,1]\to[0,\infty)$ [duplicate]
Find $$\lim_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$$if $f:[0,1]\rightarrow(0,\infty)$ is a continuous function.
My attempt:
Say $f(x)$ has a max. value $M$. ...
3
votes
5answers
197 views
Why isn't this limit equal to $0$?
$f(2)=4$, $g(2)=9$, $f'(2)=g'(2)$.
$ \displaystyle \lim_{x \to 2} \frac{ \sqrt{f(x)}-2} { \sqrt{g(x)}-2} $.
Why isn't this limit equal to $0$? Since $f$ and $g$ are differentiable at $x=2$, that ...
2
votes
1answer
29 views
Finding distributional limit
How to find $\lim_{\varepsilon\rightarrow 0+}f_{\varepsilon}$ in $D'(R)$, if $f_\varepsilon$ is defined as:
$f_\varepsilon(x)=\frac{1}{\varepsilon^3}$ for ...
1
vote
2answers
41 views
How to prove something on limit points…
$X = \mathbb R^n$. How to prove that the interior point of a subset of $X$ is also limit point?
Don't know where to start... by definition, an interior point is such if it exists an e-neighborhood ...
2
votes
1answer
32 views
Are all limits solvable without L'Hospital Rule or Series Expansion
Is it always possible to find the limit of a function without using L'Hospital Rule or Series Expansion
For example,
$$\lim_{x\to0}\frac{\tan x-x}{x^3}$$
$$\lim_{x\to0}\frac{\sin x-x}{x^3}$$
...
