Questions on the evaluation of limits.
2
votes
4answers
70 views
Finding the limit of function - irrational function
How can I find the following limit:
$$ \lim_{x \rightarrow -1 }\left(\frac{1+\sqrt[5]{x}}{1+\sqrt[7]{x}}\right)$$
0
votes
2answers
57 views
Why upper/lower limit always has only one value?
I am a beginner to calculus, and I have a simple question on limits.
Consider the function $f(x)= 1/x$ for all real $x$. Then we know that upper limit of $x$ tends to infinity is $0$. This is because ...
1
vote
1answer
38 views
Limit in a sense of distributions
How to find $\lim_{a\rightarrow\infty} f_a$ in $D'(R)$, for $a>0$, where $f_a:R\rightarrow R$ is defined by
$f_a(x)=\begin{cases}\frac{\sin{ax}}{x}&x\neq 0 \\0&x=0\end{cases}$
Thanks in ...
0
votes
1answer
26 views
how to show uniform convergence for sequence $u_n(z) = n z e^{-nz^2}$ such that $\Re[z^2] > 0$
How to show uniform convergence for $u_n(z) = n z e^{-nz^2}$ such that $\Re[z^2] > 0$
Here is my attempt letting $z = x + iy$:
$$
\begin{align*}
|u_n - 0| &= |nz e^{-nz^2}| \\
&\le n ...
14
votes
10answers
958 views
Why isn't $\lim_{x\to\infty}(1+\frac{1}{x})^{x}= 1$?
Given $\lim_{x\to\infty}(1+\frac{1}{x})^{x}$, why can't you reduce it to $\lim_{x\to\infty}(1+0)^{x}$, making the result "$1$"? Obviously, it's wrong, as the true value is $e$. Is it because the ...
1
vote
1answer
31 views
Suppose that for $a_n\geq b_n$ for all $n$. Show that $\varliminf_{n \to \infty} a_n\geq \varliminf_{n \to \infty} b_n$.
This is what I have so far:
Since $a_n\ge b_n$ for every $n$ then we have that $\inf\{a_n; n\ge k\} \ge \inf\{b_n; n\ge k\}$ for every $n$. When we take the limit as $n\rightarrow \infty$ we get ...
2
votes
1answer
24 views
I need to find the value of $a,b \in \mathbb R$ such that the given limit is true
I am given that $\lim_{x \to \infty} \sqrt[3]{8x^3+ax^2}-bx=1$ need to find the value of $a,b \in \mathbb R$ such that the given limit is true. I was able to work the whole thing out, but I have a ...
1
vote
1answer
50 views
How find this intergral $\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy$
find the value
$$\sum_{n=1}^{\infty}\int_{0}^{\pi}\int_{0}^{\pi}(xy)^{k}[\cos n(x-y)-\cos n(x+y)] \, dx \, dy,\qquad k\in N^{+}$$
my idea:
\begin{align}
...
1
vote
4answers
69 views
How to prove that $\lim_{n\to \infty} (n^k/2^n) = 0$?
I'm having a hard time trying to prove this statement.
$\lim_{n\to \infty} (n^k/2^n) = 0$
k is a positive number.
Please, help me.
Thanks in advance.
4
votes
3answers
66 views
what if take limit to negative infinity in the definition of e as a limit
By definition $$\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^n = e.$$
But what about a similar limit where $n$ tends to negative infinity, i.e, $$\lim_{n\to -\infty}\left(1+\frac{1}{n}\right)^n?$$
...
0
votes
1answer
76 views
Finding the Matrix Power of a matrix and limit
Find the matrix power, $A^k$, of
$$A=\begin{pmatrix}a & 1-a \\ b & 1-b\end{pmatrix}$$
$$D=P^{-1}AP$$
$$A^k=PD^kP^{-1}$$
I think that
$$P=\begin{pmatrix}1 & \frac{a-1}{b} \\ 1 & ...
0
votes
3answers
35 views
limit of a sequence with roots (different index)
I have to calculate the next limit:
$\lim\limits_{n \rightarrow \infty} \dfrac{2\sqrt[3]{n}-5\sqrt[5]{n^2}}{\sqrt[3]{n+1}(2-\sqrt[5]{n})}$
I've tried multiplying by the conjugate, but this give a ...
4
votes
2answers
57 views
Show $x\sqrt{n} - n \ln\left(1+\frac{x}{\sqrt{n}}\right) \to \frac{x^2}{2}$
With $x > 0$, show
$$
L=\lim_{n \to \infty} x\sqrt{n} - n \ln\left(1+\frac{x}{\sqrt{n}}\right) = \frac{x^2}{2}.
$$
I tried to write
$$
x\sqrt{n}=\ln \left( e^{x\sqrt{n}} \right),
$$
so that
$$
L ...
3
votes
4answers
100 views
How to prove that $\displaystyle \lim_{n \to \infty} \sqrt[n]{n}=1$? [duplicate]
I have seen this fact thrown around a lot and never really stopped to prove it; plugging in a few values convinced me of its truth. But I would like to see the result proved. To be clear, this is not ...
0
votes
1answer
62 views
Lim sequence $\neq$ lim subsequence
Let $\{x_n\}$ be a sequence and $\{y_k\}$ be a subsequence of $\{x_n\}$. Can you show me an example of $\{x_n\}$ such that $\displaystyle \lim_{k \to +\infty} y_k= +\infty$, but $\displaystyle \lim_{n ...
0
votes
2answers
37 views
Computing the $\lim_{h\rightarrow0}\frac{2^{2+h}-2^2}{(2+h)-2}$
I'm trying to use this function to compute the derivative.
$$\lim_{h\rightarrow0}\frac{f(x+h)-f(x)}{(x+h)-x}$$
But I'm stuck when I attempt to find the derivative of $f(2)$ for $f(x) =x^2$
The power ...
4
votes
4answers
88 views
Limit with Integral in it
$$\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 $$
...
0
votes
2answers
30 views
Investigate monotony, bound and convergence
I'm doing an exercise in the college, and I ran into a doubt: a friend says me that I've made a mistake, but I can't find it.
The exercise asks: "Investigate if the sequence $2^n\over{(n+1)!}$ is ...
3
votes
6answers
112 views
Determine the limiting behaviour of $\lim_{x \to \infty}{\frac{\sqrt{x^4+1}}{\sqrt[3]{x^6+1}}}$
Determine the limiting behaviour of $\lim_{x \to \infty}{\dfrac{\sqrt{x^4+1}}{\sqrt[3]{x^6+1}}}$
Used L'Hopitals to get $\;\dfrac{(x^6+1)^{\frac{2}{3}}}{x^2 \sqrt{x^4+1}}$ but not sure what more i ...
3
votes
5answers
472 views
Is the limit not infinity?
Is the limit of this not infinity? No matter what the value of p is? Or is there a way to simplify that fraction?
$$\lim_{k \to \infty} 2^{p}\left(\frac{k}{k+1}\right)^k$$
1
vote
3answers
33 views
Indeterminate powers and limits
The question is $\lim\limits_{x\rightarrow 0^+} x^{8 \sin(x)}$. It says, use L'Hospital's rule if necessary. Are there other methods to solve this? L'Hospital's rule would be complicated to evaluate, ...
1
vote
1answer
32 views
1
vote
1answer
63 views
What does this limit indicate?
$$\lim_{x\rightarrow\infty} \zeta(x)-\zeta(x)^{-1}-\zeta(x)^2 = -1$$
What does this limit indicate?
1
vote
3answers
66 views
Show that the sequence $(x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.
Show that the sequence $\displaystyle (x_n)=\left( \sum_{i=1}^n\frac 1 i\right)$ diverge by epsilon delta definition.
I'm not familiar with proving divergent sequence. Do anyone have any des? ...
1
vote
1answer
51 views
infinite sum limit how to find the following
Hi what is the limit of the following sum:
$$\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
76 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
237 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
65 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
149 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
48 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
68 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
53 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}?$$
I used Hopital's rule, but it kept getting more complicated!
1
vote
4answers
86 views
we need to show it is discontinuous at x≠0
can any one just explain to me to me answer
Q)
$$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 ...
