Questions on the evaluation of limits.
1
vote
4answers
60 views
How do I find the limit $\lim_{x \to 1} \frac{1}{1 - x} - \frac{3}{1 - x^3}$
How do I calculate the following limit? $$\lim_{x \to 1} \frac{1}{1 - x} - \frac{3}{1 - x^3}$$
I already transformed this into
$$\lim_{x \to 1} \frac{(1-x)(1-x)(x+2)}{(1-x)(1-x^3)} = ...
0
votes
1answer
16 views
Comparison between the limits of two real functions
I know
If $f:D(\subset\mathbb R)\to\mathbb R,c$ is a limit point of $D,$ and $f(x)\ge(\text{resp.}\le)~a~\forall~x\in D-\{c\},$ then $\displaystyle\lim_{x\to c}f(x)\ge(\text{resp.}\le)~a.$ (Provided ...
5
votes
1answer
97 views
The value of a limit of a power series: $\lim\limits_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$
What is the answer to the following limit of a power series?
$$\lim_{x\rightarrow +\infty} \sum_{k=1}^\infty (-1)^k \left(\frac{x}{k} \right)^k$$
3
votes
4answers
82 views
Infinite series and its upper and lower limit.
I am learning analysis on my own and I am puzzled with the following question.
Consider the series $$\frac{1}{2}+\frac{1}{3}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{2^3}+\frac{1}{3^3}+\cdots$$ ...
5
votes
1answer
67 views
Calculate the limit of two interrelated sequences?
I'm given two sequences:
$$a_{n+1}=\frac{1+a_n+a_nb_n}{b_n},b_{n+1}=\frac{1+b_n+a_nb_n}{a_n}$$
as well as an initial condition $a_1=1$, $b_1=2$, and am told to find: $\displaystyle ...
3
votes
2answers
70 views
How to evaluate $\lim_{x\to 0} (1+2x)^{1/x}$
Good night guys! I'm having some trouble with this:
$$\lim_{x\to 0} (1+2x)^{1/x}$$
I know that $\lim_{x\to\infty} (1 + 1/x)^x = e$
but I don't know if i should take $h=1/(2x)$ or $h=1/x$
Can ...
1
vote
2answers
44 views
confused about the limit of a trigonometric function
I am trying to calculate the limit of the following function for general $a$:
$$\lim_{x\to a}[\cos(2 \pi x)-\sin(2 \pi x) \cot(\frac{\pi x}{a})]$$
I was believing this is infinite. But Mathematica ...
6
votes
2answers
53 views
Let $a_n=n\sin(π/n)$ ,$n=1,2,… $.Then $\lim a_n=?$
Let $a_n=n\sin(π/n)$ ,$n=1,2,....$ .Then $\lim a_n$=
$0$
$π$
$1$
$∞$
how can I do this?I am totally stuck.
0
votes
1answer
17 views
Approximating sums of powers of integers: why does $\sum_{i=1}^n i^r \div \frac{n^{r+1}}{r+1} \to 1$ as $n \to \infty$?
I know there are exact formulas for sums of integer powers of integers ($\sum_{i=1}^n i^r$ with $r \in \mathbb{N}$), but I was interested in approximating them. One way occured to me through a ...
1
vote
1answer
41 views
Having trouble understanding upper and lower limits.
I am seriously having trouble understanding the meaning of upper and lower limits. Can someone give me easy-to-follow examples and explanations for the following ?
Def: Let $\{s_n\}$ be ase ...
-1
votes
1answer
49 views
1
vote
3answers
37 views
Finding the limit, multiplication by the conjugate
I need to find $$\lim_{x\to 1} \frac{2-\sqrt{3+x}}{x-1}$$
I tried and tried... friends of mine tried as well and we don't know how to get out of:
$$\lim_{x\to 1} \frac{x+1}{(x-1)(2+\sqrt{3+x})}$$
...
1
vote
2answers
92 views
How are these two equivalent?
$$\frac{\ln(e^x+x)}{x}=\frac{e^x+1}{e^x+x}$$
I see that they did something to get rid of the natural log. I couldn't find any properties that would allow me to do this. I also think that they raised ...
1
vote
1answer
42 views
REVISTED$^1$ - Order: Modular Arithmetic
Relevant Literature:
Question:
Observe that $2^{10}=1024≡−1 \pmod{25}$.Find the order of $2$ modulo $25$.
Thoughts:
Direct answers are OK, but I'd like to know if I'm right that what I'm really ...
5
votes
3answers
135 views
Find the limit without use of L'Hôpital or Taylor series: $\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2 x}\right)$
Find the limit without the use of L'Hôpital's rule or Taylor series
$$\lim \limits_{x\rightarrow 0} \left(\frac{1}{x^2}-\frac{1}{\sin^2x}\right)$$
2
votes
2answers
54 views
How to evaluate limiting value of sums of a specific type
We know that if $f$ is integrable in (0,1) then
$$
\lim_{n \to \infty} \frac{1}{n}\sum_{k=1}^{n}f(k/n) = \int_{0}^{1}f(x)dx.
$$
Recently I found the following sum
$$
\lim_{n \to \infty} ...
0
votes
2answers
57 views
$\int_0^\infty |2 \arctan\sqrt{x} - \pi {x^p}/(1+x^p)| \mathrm dx $
Let $a(x) = 2 \arctan\sqrt{x}$ and $b(x) = \pi {x^p}/(1+x^p)$. I'm trying to evaluate:
$$\int_0^\infty |a(x) - b(x)| \mathrm dx $$
I've figured out I need the integrals:
$$\int_0^1 a(x)-b(x) dx = ...
2
votes
1answer
39 views
Total area of squares.
We have a square whose length is $1$ unit. Every time we rotate by $\theta$ and scale the square such as you see in the image. Does the total area of squares converge if $\theta $ goes to $0$?
3
votes
3answers
67 views
How to find $\;\lim_{x\to 0} \frac{x}{2-\sqrt{4}-2x}\;$ without using L'Hôpital's rule?
Well, I'm studying calculus and doing some exercises. First of all, from the answers that were given by my teacher the result of this limit should be $4$. I'm beginning my calculus class now but I've ...
0
votes
2answers
72 views
How to calculate $\lim_{n \to \infty}\frac{r^n}{n}$
I'm a bit rusty on calculus and I'm not able to solve this rather simple limit:
$$\lim_{n \to \infty}\frac{r^n}{n}$$
In my case $r = -1$, and "just by looking at it" I'd guess that for $\left|r\right| ...
1
vote
2answers
36 views
Choice of numbers in calculating $\lim_{x\rightarrow 0}\frac{e^{2x}-1}{\sin 3x}$
I'm studying the limits to $\lim_{x\rightarrow 0}\frac{e^{2x}-1}{\sin 3x}$. According to my text book, I should calculate it like this:
$$\frac{e^{2x}-1}{\sin 3x} = \frac{e^{2x}-1}{2x} \times ...
0
votes
2answers
26 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
30 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
1answer
27 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
46 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
50 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
votes
4answers
60 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
$$\lim_{t\rightarrow1^+}\frac{\pi\cos(\pi t)}{\frac{-\pi\sin(\pi ...
2
votes
2answers
67 views
How to calculate $\lim\limits_{x\to1^+}\frac{1}{(x-1)^2} \int\limits_{1}^{x} \sqrt{1+\cos(\pi t)}\,\mathrm dt$
Can anyone help me by calculating this limit?
I know that I need L'Hôpital but how?
$$
\lim_{x \to 1^+}\frac{1}{(x-1)^2} \int_{1}^{x} \sqrt{1+\cos(\pi t)} \,\mathrm dt
$$
Thank you very much!!
1
vote
1answer
29 views
Differentiation - Limits Equal (Possibly MVT, Rolle's or L'Hopital)
Got a quick question from a past exam paper.
If $f:R \rightarrow R$ is differentiable, and $f$ is such that $\lim_{ x\rightarrow \infty } f(x)=\lim_{ x \rightarrow -\infty} f(x)=0$ and there is a ...
1
vote
2answers
49 views
$\lim_{x\to 0} \frac{\sin(t\sqrt{x^2-k^2})}{\sqrt{x^2-k^2}}=?$
$\lim_{x\to 0} \frac{\sin(t\sqrt{x^2-k^2})}{\sqrt{x^2-k^2}}=?$
Anyone can help, please.
Does is equal to $t$?
-1
votes
0answers
63 views
What is $\lim\limits_{|a| \to \infty} \int_0^1 (G(x) - a_0 - a_1x)^2\,dx$?
Assume the integral of g from 0 to 1 is a finite #.
$$\lim_{|a| \to \infty} f(a) = \lim_{|a| \to \infty}\int_0^1 (G(x) - a_0 - a_1x)^2\,dx$$
$a= [a_0, a_1]$, as $|a| \to \infty$, we have $a_0^2 + ...
0
votes
1answer
54 views
Probability of two random n-digit numbers dividing each other
Let $n$ be a positive integer. Suppose $a$ and $b$ are randomly (and independently) chosen two $n$-digit positive integers which consist of digits 1, 2, 3, ..., 9. (So in particular neither $a$ nor ...
1
vote
2answers
51 views
Convergence of these series
$$\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
votes
4answers
64 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.
$$\lim_{x\to +\infty} ...
0
votes
1answer
57 views
Compute limit of the sequence given by $x_1=1$, $x_{n+1}^2=\frac{x_n+3}{2}$
Let $\left(x_n\right)$ be a real sequence such that $x_1=1,x_{n+1}^2=\dfrac{x_n+3}{2},\forall n\geq 1$.Compute $\lim_{n\to+\infty} 3^n\sqrt{\dfrac{9}{4}-x_n^2}$
2
votes
1answer
77 views
Does $\lim\limits_{n\to\infty}\frac{\sum_{i=1}^{n^2} 1}{\sum_{i=1}^n i}$ result into 2?
How does the following limit $$\lim\limits_{n\to\infty}\frac{\displaystyle\sum_{i=1}^{n^2} 1}{\displaystyle\sum_{i=1}^n i}$$ result into 2?
Something like:
...
2
votes
1answer
27 views
For the following monic polynomial,$f$ of even degree how to prove that that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
For any monic polynomial $f \in \mathbb {Q[x]}$ of even degree,how to prove, there exists polynomial $g \in \mathbb {Q[x]}$ such that $lim_{|x|\to\infty }(\sqrt {f(x)}-g(x))=0$
0
votes
3answers
38 views
Evaluating $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$
I am trying to evaluate $\int_0^{\infty}\frac{2}{x^2-8x+15}dx$. Factoring and using partial fraction decomposition, I have found that the indefinite integral is:
$$\ln{|x-5|}-\ln{|x-3|} + C$$
But ...
5
votes
4answers
112 views
Evaluating $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$
Could you help me evaluate $\lim _{n \rightarrow \infty} (2n+1) \int_0 ^{1} x^n e^x dx$?
I've calculated that the recurrence relation for this integral is:
$\int_0 ^{1} x^n e^x dx = x^ne^x | ^{1} ...
4
votes
3answers
96 views
Proving if $f(x)$ is differentiable at $x = x_0$ then $f(x)$ is continuous at $x = x_0$.
Please, see if I made some mistake in the proof below. I mention some theorems in the proof:
The condition to $f(x)$ be continuous at $x=x_0$ is $\lim\limits_{x\to x_0} f(x)=f(x_0)$.
(1) If $f(x)$ ...
5
votes
5answers
116 views
Limit as $x$ approaches $1$ from the right of $\frac{1}{\ln x}-\frac{1}{x-1}$
$$
\lim_{x\rightarrow 1^+}\;\frac{1}{\ln x}-\frac{1}{x-1}
$$
So I would just like to know how to begin to solve this limit, or what topic does this problem fall under so that I can search for ...
6
votes
2answers
135 views
High school contest question
Some work on it reveals the possibility of using gamma function. Is there any easy way to compute it?
$$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$
2
votes
1answer
34 views
Question about limits with variable on exponent
So I have to find the following limit $$\lim_{n\to\infty}\left(1+\frac{2}{n}\right)^{1/n}.$$I said that this is ...
10
votes
6answers
129 views
A limit on binomial coefficients
Let $$x_n=\frac{1}{n^2}\sum_{k=0}^n \ln\left(n\atop k\right).$$ Find the limit of $x_n$.
What I can do is just use Stolz formula. But I could not proceed.
1
vote
1answer
25 views
$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0?$ for $f\in L^{p}$, $p \in [1,\infty)$
For $f\in L^{p}$, $p \in [1,\infty)$
we want to prove:
$$\lim_{y \to \infty}\int_{R}f(x-t)\frac{t}{t^2 +y^2}dt=0$$
I'm not sure whether we can exchange the limit and the integral, cuz I cannot find ...
4
votes
3answers
106 views
Limit. $\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$
I want to compute $$\lim_{x \to \infty}{\sin{\sqrt{x+1}}-\sin{\sqrt{x}}}.$$
Is it OK how I want to do?
...
3
votes
2answers
71 views
Limit: $\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$
How can I find the following limit?
$$\lim_{x \to 0}{[1+\ln{(1+x)}+\ln(1+2x)+\ldots+\ln(1+nx)]}^{1/x}.$$
It's a limit of type $\displaystyle 1^{\infty}$ and if I note with $\displaystyle ...
2
votes
2answers
57 views
Proof f(x) is continuous given $x$ rational and irrational.
How can I resolve the task below:
Given $f(x)=
\begin{cases}
x, &x\in \mathbb{Q}\text{ }\\
1-x, &x\notin \mathbb{Q}\text{ (irrational)}
\end{cases}$, $0 \leq x \leq 1$.
Show $f(x)$ is ...
4
votes
8answers
141 views
Limit of $\lim_{x \to 0}\left (x\cdot \sin\left(\dfrac{1}{x}\right)\right)$ is $0$ or $1$?
WolframAlpha says $\lim_{x \to 0} x\cdot \sin\left(\dfrac{1}{x}\right)=0$ but I've found it $1$ as below:
$$
\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 $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?
