Questions on the evaluation of limits.
0
votes
1answer
62 views
Lim sequence $\neq$ lim subsequence
Let $\{x_n\}$ be a sequence and $\{y_k\}$ be a subsequence of $\{x_n\}$. Can you show me an example of $\{x_n\}$ such that $\displaystyle \lim_{k \to +\infty} y_k= +\infty$, but $\displaystyle \lim_{n ...
0
votes
2answers
37 views
Computing the $\lim_{h\rightarrow0}\frac{2^{2+h}-2^2}{(2+h)-2}$
I'm trying to use this function to compute the derivative.
$$\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{(x+h)-x}$$
But I'm stuck when I attempt to find the derivative of $f(2)$ for $f(x) =x^2$
The power ...
4
votes
4answers
90 views
Limit with Integral in it
$$\lim_{x \to 0} \dfrac{\displaystyle \int_0^x \sin \left(\pi t^2/2\right) dt}{x^3}$$
I am having trouble trying to figure out how to compute the limit.
Do I have to take the integral first and then ...
1
vote
1answer
47 views
How to prove that $n^k = O(2^n)$
I'm having issues trying to prove this.
The Big Oh definition is: f(n) = O(g(n)) if exists a real constant $c > 0$ and $n_0 \in \Bbb N $ in such a way that for all $n \ge n_0$ we have f(n) $\le$ ...
1
vote
3answers
67 views
Finding the limit $\lim_{n\to \infty}\int^n_0 e^{-\lambda x}\mathrm dx$
Find the following limit:
$$\lim_{n\to \infty}\int^n_0 e^{-\lambda x}\mathrm dx$$
for all $\lambda>0$.
2
votes
2answers
55 views
Exponential Limit
What is $\lim_{n\rightarrow\infty}(1 + \frac{3}{n})^n$?
I'm a little confused on this limit. The $\frac{3}{n}$ part gets smaller as $n$ gets bigger, so it would really just come out to $1^n$, right?
1
vote
1answer
44 views
Prove that for every $p>0$, $\lim_{n\rightarrow∞}\int_n^{n+p}{\sin x\over x} = 0$
Got stuck with this question:
Prove that for every $p>0$, $\displaystyle \lim \limits_{n\rightarrow∞}\int_n^{n+p}{\sin (x)\over x} = 0$.
Thanks in advance for any help!
3
votes
2answers
58 views
Proving $\lim_{x \rightarrow{1}^{-}} \int_{-x}^{x}\frac { f(t)}{\sqrt { 1-{ t }^{ 2 } } }dt$ exists
Let $\textit{f} :[-1,1] \rightarrow \mathbb{R}$ continuous on $[-1,1]$
I need to prove that $$\lim_{x \rightarrow{1}^{-}} \int_{-x}^{x}\frac { f(t)}{\sqrt { 1-{ t }^{ 2 } }}dt$$
exists
But I have ...
2
votes
2answers
53 views
Confusion over a limit. Different ways of solving give different answers?
Qn: If it is given that
$$
\lim_{x\to\infty} \frac{x^2 - x - 2}{x + 1} - ax - b = 1
$$
then a and b must be?
Now, I tried doing this by 2 methods.
Method 1:
$$ \frac{x^2 - x - 2}{x + 1} - ax - b $$
...
0
votes
2answers
31 views
Investigate monotony, bound and convergence
I'm doing an exercise in the college, and I ran into a doubt: a friend says me that I've made a mistake, but I can't find it.
The exercise asks: "Investigate if the sequence $2^n\over{(n+1)!}$ is ...
3
votes
6answers
113 views
Determine the limiting behaviour of $\lim_{x \to \infty}{\frac{\sqrt{x^4+1}}{\sqrt[3]{x^6+1}}}$
Determine the limiting behaviour of $\lim_{x \to \infty}{\dfrac{\sqrt{x^4+1}}{\sqrt[3]{x^6+1}}}$
Used L'Hopitals to get $\;\dfrac{(x^6+1)^{\frac{2}{3}}}{x^2 \sqrt{x^4+1}}$ but not sure what more i ...
3
votes
5answers
473 views
Is the limit not infinity?
Is the limit of this not infinity? No matter what the value of p is? Or is there a way to simplify that fraction?
$$\lim_{k \to \infty} 2^{p}\left(\frac{k}{k+1}\right)^k$$
1
vote
3answers
33 views
Indeterminate powers and limits
The question is $\lim\limits_{x\rightarrow 0^+} x^{8 \sin(x)}$. It says, use L'Hospital's rule if necessary. Are there other methods to solve this? L'Hospital's rule would be complicated to evaluate, ...
1
vote
1answer
32 views
1
vote
1answer
64 views
What does this limit indicate?
$$\lim_{x\rightarrow\infty} \zeta(x)-\zeta(x)^{-1}-\zeta(x)^2 = -1$$
What does this limit indicate?
1
vote
3answers
68 views
Show that the sequence $(x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.
Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.
I'm not familiar with proving divergent sequence. Do anyone have any des? ...
1
vote
1answer
52 views
infinite sum limit how to find the following
Hi what is the limit of the following sum:
$$\lim \limits_{n\rightarrow\infty}\frac{2}{n^2}\sum\limits_{j=0}^{n-1}\sum\limits_{k=j+1}^{n-1}\frac{k}{n}$$
Thanks a lot!
2
votes
2answers
48 views
Does the equality $\lim_{x\rightarrow\infty} f(g(x)) = f(\lim_{x\rightarrow\infty} g(x))$ hold?
Does the equality $\lim_{x\rightarrow\infty} f(g(x)) = f(\lim_{x\rightarrow\infty} g(x))$ hold? If it is not always true, what is the condition that makes the equality hold?
1
vote
4answers
52 views
L'Hospital's Rule and indeterminate powers
What is $\displaystyle \lim_{x\to\infty}\left(\frac{17x}{17x+9}\right)^{3x}$?
I tried to solve this problem and could not understand this.
I know that it is an exponential equation of the type ...
3
votes
1answer
109 views
Can't prove this limit of complex numbers from a paper
Okay so I found in a paper, marked as "simple exercise", the following thing: for $z,b \in \mathbb{C}$,
$$\lim_{b\to0} \frac{1}{z-b} + \frac{1}{2b} - ...
0
votes
2answers
78 views
Calculate the limit $\lim\limits_{n\to\infty}\frac{\sqrt[n]{|x-1|}+1}{(x+1)^n+1}$
How can I compute the following limit?
$$\lim_{n\to\infty}\frac{\sqrt[n]{|x-1|}+1}{(x+1)^n+1}$$
5
votes
2answers
115 views
How to calculate $ \lim_{n\to\infty} (2^n+3^n+\cdots+n^n)^{1/n}/n ?$
I need help in calculating the following limit.
$$\lim_{n\to\infty}\frac{\sqrt[n]{2^n+3^n+\cdots +n^n}}{n}$$
5
votes
4answers
262 views
What is a simple example of a limit in the real world?
This morning, I read Wikipedia's informal definition of a limit:
Informally, a function f assigns an output f(x) to every input x. The
function has a limit L at an input p if f(x) is "close" to ...
0
votes
1answer
67 views
Prove that the sequence $(\frac{n^2}{n!})$ converge using epsilon delta definition.
Prove that the sequence $(\frac{n^2}{n!})$ converge using epsilon delta definition.
I have never seen limit involving factorial, do anyone has any ideas? Thank you.
0
votes
2answers
49 views
Find a convergent function in metric space
Let $C[−1, 1]$ be the space of continuous functions equipped with the metric $p(f,g) = \max\{|f(x)−g(x)| \mid x \in [−1, 1]\}$.
Then the sequence of functions $(f_n):[−1,1]\rightarrow \mathbb{R}$ ...
1
vote
0answers
52 views
2
votes
2answers
64 views
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$
$x_n$ is the $n$'th positive solution to $x=\tan(x)$. Find $\lim_{n\to\infty}\left(x_n-x_{n-1}\right)$.
3
votes
6answers
152 views
Strange behavior of $\lim_{x\to0}\frac{\sin\left(x\sin\left(\frac1x\right)\right)}{x\sin\left(\frac1x\right)}$
Alright, scratch everything below the line. Let me present one cohesive question not marred by repeated edits.
The limit $\lim_{x\to a}f(x)=L$ exists iff for every $\epsilon>0$ there is a ...
2
votes
1answer
51 views
Limit of Binomial distribution
In showing us that Binomial distribution: $$B_{N,p}(n) := \binom {N}{n} p^n(1-p)^{N-n}$$ tends to Poisson's: $$P_ \lambda (n) = \dfrac {\lambda ^n}{n!}e^{-\lambda}$$where I guess lambda should be ...
1
vote
2answers
69 views
Removing the Indeterminate Form of a Limit involving Natural Logs where X approaches 1
It's pretty sad, but I've been working on this math problem for a couple of hours, now. Still scouring my Calculus textbook (Calculus Concepts and Contexts by James Stewart,) class notes, and Math ...
3
votes
3answers
55 views
proving that the following limit exist
How can I prove that the following limit exist?
$$
\mathop {\lim }\limits_{x,y \to 0} \frac{{x^2 + y^4 }}
{{\left| x \right| + 3\left| y \right|}}
$$
I tried a lot of tricks. At least assuming that ...
1
vote
1answer
29 views
Quick question about limits.
Sometimes, when we take limits, especially for roots and ratio tests, we define
lim of sup(a_n). Is this only because when we take the limit of sup we don't have to worry about the existence of the ...
1
vote
1answer
35 views
Existence of a continuous real-valued function in two real variables
Does there exist a continuous function $F:\mathbb{R}^{2}\to\mathbb{R}$ with $D(F)=\mathbb{R}^{2}$, that is $|F(x,y)|<+\infty$ for all $(x,y)\in\mathbb{R}^{2}$, such that
$$\lim_{|x-y|\to 0} ...
0
votes
1answer
34 views
Finding jump conditions
I have the equation:
$ \dfrac{1}{r^2}\dfrac{d}{dr}\left( r^2 \dfrac{d y(r,t)}{dr} \right) - \dfrac{ y(r,t)}{r^2} = S \delta(r-a(t)) $
where S is a function of t alone, and I want to find jump ...
2
votes
3answers
68 views
Tricky Limit question
What is
$$\lim_{t \rightarrow 0} \frac{e^{t^3}-1-t^3}{\sin(t^2)-t^2}?$$
I used Hopital's rule, but it kept getting more complicated!
1
vote
4answers
87 views
we need to show it is discontinuous at x≠0
can any one just explain to me to me answer
Q)
$$f(x)=\begin{cases}
x &\text{if }x\in \mathbb{Q} \\
0 &\text{if }x\in \mathbb{R}\setminus\mathbb{Q}
\end{cases}$$
we need to show it is ...
0
votes
1answer
47 views
Problem on Yukawa Potential
One definition of the Yukawa potential on $R^n$ is the solution $G$ in the sense of distributions to $(-\Delta + \mu^2)G = \delta$. This 'green's function' is given by
\begin{align*}
G(x) = ...
0
votes
0answers
36 views
Infimum of a multivariable function
How to find the infimum of following equation over y.
$$f(x,y)=x^T Ax+x^T By+y^T Bx+y^T Cy$$
$$inf_y f(x,y)=?$$
where A and C are symetric matrices
18
votes
3answers
377 views
When L'Hôpital's Rule Fails
I was discussing L'Hôpital's Rule with a Calculus I student earlier today. I mentioned that if the limit from LH doesn't exist, then L'Hôpital's Rule tells us nothing about the original Limit.
A ...
0
votes
1answer
39 views
Is there any way of making this true: $ 0 \leq \lim\limits_{(x,y)\rightarrow(\infty,0)} xy \leq \epsilon $
Is there any conditions that make the following sentence exist?
$$
0 \leq \lim_{(x,y)\rightarrow(\infty,0)} xy \leq \epsilon
$$
4
votes
2answers
80 views
Let $f>0$ differentiable in $[0,\infty)$. Assume $\lim \limits_{x \to \infty} (\log\circ f)^\prime(x) < 0$. Show that $\int_0^\infty f$ converges.
So what I gathered from the givens about $f$, since $(\log\circ f)^\prime(x)=\frac{f^\prime(x)}{f(x)}$ it would mean that far enough, $f^\prime(x)<0$. I don't know how to go about this from here.
...
1
vote
2answers
76 views
The mathematical and technical approach to a limit of a sum of a sequence
In Calculus, what is the most preferred mathematical and technical way to approach a limit of a sum of a sequence?
Take for example:
$$ \lim_{n \to \infty} {\frac{1}{n} \sum_{k=1}^{n} {\ln(1 + ...
0
votes
1answer
79 views
Evaluating limits with variable exponent and an exponent function with fraction
I face a problem when trying to evaluate the following limits.
$\lim\limits_{x\rightarrow 0} ((\dfrac{e^2}{x} +3)/4)^4x$
Please help. Thank you.
0
votes
0answers
61 views
Why is $ \lim_{x \to \infty} {\arctan(x)} = \frac{\pi}{2} $ [duplicate]
Why is:
$$ \lim_{x \to \infty} {\arctan(x)} = \frac{\pi}{2} $$
Does it come from the way arctan behaves in the positive side of X?
1
vote
1answer
52 views
Show that $\lim(\frac 1 n-\frac 1 {n+1})=0$ using epsilon-delta definition.
Show that $\lim(\frac 1 n-\frac 1 {n+1})=0$ using epsilon-delta definition.
I have to show that if $n,N\in\mathbb N$ and $n\geq N$, then $\frac 1 n-\frac 1 {n+1}\leq \frac 1 N-\frac 1 {N+1}$, but ...
0
votes
3answers
45 views
Limit of Exponential Integral Function
I want to ask how to prove whether the following limit is corrent
$$
\lim_{x \to 0} \left[ {x{e^x}{E_1}( x )} \right] = 0,
$$
with
$\displaystyle {E_1}( x ) = \int_x^\infty \frac{e^{ - t}}{t}dt$. I ...
1
vote
2answers
49 views
Prove or disprove a result for a double sequence.
Suppose that a double sequence $\{a_{n,k}\}=\left\{\frac{1}{n^{\frac{k-1}{k}}}\right\}$. Prove or disprove $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k}a_{n,k}=0$.
0
votes
0answers
16 views
Help with a limit
Can someone help me evaluate $$\lim_{x \to 1}\frac{x^{\omega}(1-x^{a})^{\omega}}{a}\sum_{t=1}^a\frac{e^{2\pi i t(\omega-b)/a}}{(1-xe^{2\pi i t/a})^\omega}$$
For arbitrary integers ($\omega$ , $b$ , ...
1
vote
4answers
92 views
$\lim_{x\rightarrow\infty}\sin(x)$?
In physics I came across these kind of equations when I am trying to find the asymptotic behaviour of some function.
Can anyone explain if there is any sense in talking about $\sin(x)$ or $\cos(x)$ ...
1
vote
5answers
192 views
Prove that $\cos(x)$ doesn't have a limit as $x$ approaches infinity.
I've been working on this one for quite a long time now.
I have to prove that $\cos(x)$ has no limit as $x$ approaches infinity.
Let $\epsilon>o$ and M be any number greater than 0, so that for ...
