Questions on the evaluation of limits.
5
votes
4answers
444 views
Evaluate $\lim\limits_{n\to\infty}\frac{\sum_{k=1}^n k^m}{n^{m+1}}$
By considering:
$$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^1}{n^{2}} = \frac 1 2$$
$$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^2}{n^{3}} = \frac 1 3$$
$$\lim_{n\to\infty}\frac{\sum_{k=1}^n k^3}{n^{4}} = ...
5
votes
2answers
161 views
A trigonometric-integral inequality
This problem comes from a discussion with one of my friends:
Prove that: $$\displaystyle \lim_{n \to \infty}\int_{1}^{n}{\sin (x)\sin(x^2)}\,{\mathrm dx}< \lim_{n \to ...
1
vote
2answers
80 views
$\lim_{n\rightarrow \infty} (n+1)\cdot x^{n\cdot n!} < 1$
If $\vert x \vert < 1$ then I want to show $\lim_{n\rightarrow \infty}
(n+1)\cdot x^{n\cdot n!} < 1$
It makes perfectly sense in my world, because the factor $x^{n\cdot n!}$ is smaller ...
5
votes
2answers
160 views
Alternative proof of the limitof the quotient of two sums.
I found the following problem by Apostol: Let $a \in \Bbb R$ and $s_n(a)=\sum\limits_{k=1}^n k^a$. Find
$$\lim_{n\to +\infty} \frac{s_n(a+1)}{ns_n(a)}$$
After some struggling and helpless ideas I ...
0
votes
3answers
58 views
prove the limit in a formal way
I want to show the below statement:
$\lim_{n\rightarrow \infty} \dfrac{n+1}{2^{n\cdot n!}} = 0$
I can see that it is true, because the part $2^{n\cdot n!}$ will be greater than $n+1$, when ...
4
votes
1answer
224 views
The limit of an infinite sum …
Calculate the following limit:
$$\lim_{n\to\infty} \left(\sum_{k=0}^n \frac{{(1+k)}^{k}-{k}^{k}}{k!}\right)^{1/n} $$
First of all, i'm just looking for any helping hint that will alow me to solve
...
3
votes
1answer
247 views
Binomial fraction sum to infinity
Compute the limit:
$$\lim_{n\to\infty} \sum_{k=0}^n \frac {\dbinom{n}{k}}{\dbinom{2n-1}{k}}$$
Here i tried to give some k values to the sum hoping to see a possible pattern,
but i didn't figure out ...
4
votes
1answer
169 views
How to calculate the limit of a sequence?
Calculate the limit of the sequence
$$\lim_{n\rightarrow\infty}\ a_n$$
$$a_n=\left(1-\frac{1}{2^2}\right)\left(1-\frac{1}{3^2}\right)\ldots\left(1-\frac{1}{n^2}\right), n\geqslant2 $$
Here is what ...
10
votes
3answers
211 views
$\cos(x)$ and $\arccos(x)$ couple limit
Find the value of the following limit:
$$\lim_{n\to\infty} \frac {\cos 1 \cdot \arccos \frac{1}{n}+\cos\frac
{1}{2} \cdot \arccos \frac{1}{(n-1)}+ \cdots +\cos \frac{1}{n} \cdot
...
-1
votes
3answers
107 views
Calculate the limit of the sequence $\lim_{n\rightarrow\infty}\ a_n, n\geqslant1 $
Calculate the limit of the sequence
$$\lim_{n\rightarrow\infty}\ a_n, n\geqslant1 $$
knowing that
$$\ a_n = \frac{3^n}{n!},n\geqslant1$$
Choose the right answer:
a) $1$
b) $0$
c) $3$
d) ...
1
vote
0answers
191 views
Polynomial limit to infinity
Let be a polynomial with real coefficients. Calculate the value of this limit:
$$\lim_{n \rightarrow \infty} |P(1)...P(n+1)|^ \frac1{n+1}-|P(1)...P(n)|^ \frac1{n} $$
3
votes
5answers
164 views
Something is wrong with this proof, limits
Could someone please tell me what is wrong with this proof?
Show that $\lim\limits_{(x,y) \to (0,0)} \dfrac{xy^3}{x^4 + 3y^4}$ does not have a limit or show that it does and find the limit.
I ...
2
votes
3answers
55 views
Big-$\mathcal{O}$ bounding of sums of logarithmic functions
I am reading a text which states that $$\sum \limits_{n \leq X} \left(\log X - \log n \right) = \mathcal{O}(X)$$ I can't quite see why this is true, though I can certainly believe it. Could anyone ...
0
votes
0answers
135 views
How to calculate partial derivatives of $f(x+iy)=x^2-y^2 + 5xi$ using limits
Let $f(x+iy)=x^2-y^2 + 5xi$. So hence $u(x,y)=x^2-y^2$ and $v(x,y)=5x$
In my notes it calculated $\frac{\partial u}{\partial x}$ at $0$ as follows:
$\frac{\partial u}{\partial ...
17
votes
3answers
818 views
$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$ Evaluating this limit
Evaluate $$\lim_{n\to\infty} \left(\frac{1^p+2^p+3^p + \cdots + n^p}{n^p} - \frac{n}{p+1}\right)$$
5
votes
2answers
162 views
Is this sum related to the Gregory's limit?
Compute the series
$$\sum_{k=0}^{\infty}\left(\frac{1}{4k+1} - \frac{1}{4k+2}\right)$$
2
votes
2answers
90 views
Looking for some function such that $\lim\limits_{x\to\infty}f(x) \ne \infty$
I am looking for a function $f$ that is differentiable and $f'(x) \ge c \gt 0$ for all $x \in \mathbb{R}$ and $\lim\limits_{x\to\infty}f(x) \ne \infty$?
Is there such function, or am I wasting my ...
8
votes
3answers
643 views
Computing $\lim\limits_{n\to\infty} \int_{0}^{\infty}\frac{\sin(x/n)}{(1+x/n)^{n}}\, dx$
$$\lim_{n\to\infty} \int_{0}^{\infty}\frac{\sin(x/n)}{(1+x/n)^{n}}\, dx$$
I've been able to show that the integral is bounded above by 1 (several ways).
One of the simplest is just letting $u=x/n$ ...
0
votes
1answer
91 views
How to solve the limit of a succession on this particular circumstances?
Given this limit
$\displaystyle\lim_{n \to{+}\infty}{\frac{\sqrt{16n^2+3}}{(1+a_n)n+5cos n}=\frac{7}{6}}$
I need to calculate this one :
$\displaystyle\lim_{n \to{+}\infty}{a_n}$
Any ideas of how to ...
2
votes
2answers
134 views
How to find the limit of a function?
Find $$\lim\limits_{n \to \infty} \left( \frac{n^{1/3}}{2} \text{arccos} \left(\dfrac1{\sqrt{1+\frac{4}{(k(n)-1)^2}}\sqrt{1+\frac{8}{(k(n)-1)^2}}} \right) \right).$$
where $k(n) = ...
2
votes
5answers
117 views
Finding the limit of $(1-3 \cdot x)^{\frac{1}{x}}$
I am trying to find $\lim_{x \rightarrow0} (1-3 \cdot x)^{\frac{1}{x}}$
I thought about finding the limit of
$$(1-3 \cdot x)^{\frac{1}{x}}= e^{\frac{\ln(1-3 \cdot x)}{x}}$$
But that only works if ...
7
votes
3answers
601 views
Proof of the L'Hôpital Rule for $\frac{\infty}{\infty}$
I ask for the proof of the L'Hôpital rule for the indeterminate form $\frac{\infty}{\infty}$ utilizing the rule for the form $\frac{0}{0}$.
Theorem: Let $f,g:(a,b)\to \mathbb{R}$ be two ...
5
votes
5answers
386 views
Prove that if $a\in [0,1]$, then $\lim\limits_{x \to a} f(x) =0$
Supose that for any natural number $n$, $A_n$ is a finite set of numbers from $[0,1]$, and that $A_m$ and $A_n$ have no common elements if $m \neq n$, ie
$$m \neq n \Rightarrow A_n\cap ...
0
votes
1answer
105 views
Nowhere else continuous function differentiable at $x=0$?
I saw an interesting question:
Let $D(x)=\begin{cases} 1 & \text{if } x \in \mathbb{Q}, \\ 0 & \text{if } x \in \mathbb{R} \setminus \mathbb{Q}\end{cases}$
Let $f(x) = x \cdot D(x)$.
...
4
votes
6answers
280 views
Help me evaluate limit of sequence
I have this limit, and i have no idea of approach:
$$\lim_{n \rightarrow + \infty } \left(\frac{n^3}{4n-7}\right)\left(\cos\left(\frac1n\right)-1\right)$$
turns out to be of indeterminate form, how ...
0
votes
1answer
75 views
Continuity and limits
I am asked to use continuity to evaluate the limit:
$$\lim_{x\to 1} e^{x^2-x} $$
I know that evaluating both the limit and the function will produce the same value, 1, so that tells me that it is ...
2
votes
1answer
96 views
How to find limit of function
How would I find this limit?
$\lim_{n \to \infty} \frac{\sqrt{n}}{2} \bigl(\arccos(\frac{n-2}{22+n})) $
0
votes
3answers
373 views
Trigonometric limit solution
Why does the following limit equals 2:
$$\lim_{x \to 0}\frac{2x^2}{\sin^2 x}=2$$
I can't find a trigonometric conversion to get that result.
8
votes
3answers
516 views
Is the infinite root of any number equal to $1$?
I was messing around in IRB and I decided to make a $n^{th}$ root function and noticed that for very large roots of numbers, the answer always converges to $1$. It has been a while since I have done ...
3
votes
1answer
498 views
Difference in limits because of greatest-integer function
A Problem : \begin{equation}\lim_{x\to 0} \frac{\sin x}{x}\end{equation} results in the solution : 1
But the same function enclosed in a greatest integer function results in a 0
...
2
votes
2answers
251 views
Question On the Proof of The boundedness Theorem
let $f:[a,b]\rightarrow\mathbb{R}$, f continuous on $[a,b]$. I shall prove that $\exists A,B\in\mathbb{R}, \forall x\in[a,b], A\le f(x)\le B$.
Proof: Let's define $g(x)=|f(x)|$, we need to prove now ...
5
votes
3answers
196 views
Is a simplified function the same as the original?
Is a simplified function the same as the original?
Example:
Let $f(x) = \frac{ax}{x}$, and $g(x) = a$
where $a$ and $x$ are real numbers.
Does $g$ = $f$?
2
votes
3answers
113 views
Limit Evaluation
Given $a>1$ and $f:\mathbb{R}\backslash{\{0}\} \rightarrow\mathbb{R}$ defined $f(x)=a^\frac{1}{x}$
how do I show that $\lim_{x \to 0^+}f(x)=\infty$?
Also, is the following claim on sequences ...
1
vote
3answers
110 views
how to evaluate $\lim_{x \to \infty}(1+4/x)^\sqrt{x^2+1}$
$$
\lim_{x \to \infty}(1+4/x)^\sqrt{x^2+1}
$$
is like
$$
\lim_{x \to \infty}(1+1/x)^x = e
$$
I have replaced
$\sqrt{x^2+1}$ by $x$ but I haven't got the expected result ($e^4$).
1
vote
1answer
96 views
Try to evaluate $\lim_{\alpha\to1^-}\frac{1}{\alpha(\alpha-1)}\left[\frac{1}{m}\sum_{i=1}^{m}\left(\frac{y_i}{y^{'}}\right)^\alpha-1\right]$
I can't evaluate this limit.
$$\lim_{\alpha\to1^-}\frac{1}{\alpha(\alpha-1)}\left[\frac{1}{m}\sum_{i=1}^{m}\left(\frac{y_i}{y^{'}}\right)^\alpha-1\right]$$
where $y_i>0$, $y^{'}$ is the average of ...
1
vote
1answer
147 views
One sided limit of an increasing function defined on an open interval
Let $f:(a,b)\to \mathbb{R}$ be a strictly increasing function. Does the limit $\lim_{x\to a^+}f(x)$ necessarily exist and is a real number or $-\infty$? If so, is it true that $\ell=\lim_{x\to ...
2
votes
2answers
114 views
How does the definition of “limit” capture the idea that a sequence gets “closer and closer” to the limit?
The sequence $x_n=2+1/n$ certainly gets "closer and closer" to $2$ as $n$ gets "larger and larger." And we know that
$$\lim_{n\to\infty}2+1/n=2$$
What's going on here? Use the definition to ...
1
vote
1answer
466 views
list of convergent series
I wanted to know if there is an online reference I can use to find out known results about convergent series. I could not find this one, for example, on wikipedia
$\sum_{k=1}^{+\infty} ...
0
votes
2answers
48 views
Convex function with linear grow?
I'm looking for a continuous, strictly increasing, strictly convex function $f: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}_{\geq 0}$, with $f(0)=0$, and such that
$$ \lim_{x \rightarrow\infty} ...
2
votes
1answer
86 views
Help with question involving limits, bounds, inequalities
Would someone like to help with the following question?
Prove that for $n=1,2,\ldots$
(a) $5\leq (4^n+5^n)^{1/n}\leq 10$ and that $(4^n+5^n)^{1/n}$ is bounded,
(b) $(4^n+5^n)^{1/n}\geq ...
1
vote
1answer
89 views
A limit related to the Laplace constant
Let $x\in (0,1)$,
$$
M(v,x) =
\bigl(1+v^x\bigr)
\bigl(v^{1/x}-v\bigr)
+
x
\bigl(1+v^{1/x}\bigr)
\bigl(v^x-v\bigr)
$$
and let $v_0(x)$ be the root of $M(v,x)$ in $(0,1)$. As $x \rightarrow 1$ this ...
1
vote
1answer
311 views
Power rule for the limits of convergent sequences
The "Power Rule" for null sequences states that
If a null sequence of non-negative terms is raised to a positive
power, the resulting sequence is also a null sequence.
Ok, can this rule be ...
2
votes
3answers
247 views
Are undefined terms allowed in a sequence?
Is this a valid sequence? $\{\frac1{(n-3)}\}$ I.e. Can a sequence have individual terms which are undefined?
And if so, does this mean that the above sequence is unbounded (since the third term is ...
3
votes
4answers
139 views
Calculate $\lim\limits_{x\to -1}\frac{x^2+3x+2}{x^2+2x+1}$
I've just have a mathematics exam and a question was this:
Calculate the limits of $\dfrac{x^2+3x+2}{x^2+2x+1}$ when $x\text{ aproaches }-1$.
I started by dividing it using the polynomial long ...
1
vote
1answer
66 views
Question Regarding Existence of One Sided Limits
In a Calculus book, I had read the following proposition:
For a function $f:X\to \mathbb{R}$, $X\subseteq \mathbb{R}$ then $\lim\limits_{x\to x_0}f(x)$ exists if and only if $\lim\limits_{x\to ...
2
votes
5answers
217 views
Trig limit of $\lim\limits_{x\to 0}\frac{\sin{6x}}{\sin{2x}}$
$$\lim_{x\to 0}\frac{\sin{6x}}{\sin{2x}}$$
I have no idea at all on how to proceed. I am guessing there is some trig rule about manipulating these terms in some way but I can not find it in my notes.
...
4
votes
4answers
250 views
Finding $\lim\limits_{x\to 0}\frac{\sin{3x}}{x}$
I am trying to find the limit of
$$\lim_{x\to 0}\frac{\sin{3x}}{x}$$
I have no idea what I am supposed to do. I know the identity that,
$$\lim_{x\to0}\frac{\sin{x}}{x} = 1$$
but that will not be ...
2
votes
1answer
64 views
Exercise: optimization and limit
Let $\underline{a}_N:=(a_1,...,a_N)^\top \in \mathbb{R}^N$ denote a vector of lenght $N$.
Define
$$ x_N^* := \arg \min_{x \in \mathbb{R}} \left\| \underline{a}_N - x \mathbb{1}_N \right\|_p $$
where ...
0
votes
2answers
143 views
Help me with this Limit
I don't have idea how to make this limit, i read it in a math contest.
I think that is a limit that could be attacked by method of Riemann's sums.
$$\lim_{x\to 0} \int _0 ^ {x} (1- \tan (2t) ) ^ ...
15
votes
6answers
863 views
Limits of $f(x)=x-x$
It's obvious that $f(x)=x-x=0$. But what exactly happens here?
You have a function $f(x)=x-x$ and you have to calculate the limits when $x\to \infty$
This'll be like this:
$$\lim\limits_{x\to ...
