Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [limits]

Questions on the evaluation and properties of limits in the sense of analysis and related fields. For limits in the sense of category theory, use (limits-colimits) instead.

0
votes
0answers
3 views

Continuous Function with Infimum Metric Measure

For any set A ⊂ R λ, x ∈ R denote $dist(x,A)= inf_{a∈A} |x−a|$, $\lambda·A$={$\lambda$ a|a∈A} For k ∈ N, we define $g_n(x) = dist (x, 2^{−n}Z)$. consider $f_N : [0,1] → R$, $$f_N(x) = \sum_{n=0}^N ...
-1
votes
0answers
8 views

Given this transition probability matrix how do i go about computing the limiting probabilities?

[Question][1] [1]: https://i.stack.imgur.com/nV9Sa.jpg I was working on a problem to practice for an upcoming test and got stuck filling up the entries of the ...
0
votes
0answers
8 views

Calculation of Marginal Utility using limit

I am unsure about a problem in marginal utility calculation Calculate MU1(x1,x2) for $u(x1,x2)= x1^a + x2^b$ By definition, $ MU1= \lim {\delta x1\to 0} \frac {\delta U} {\delta x1} $ $= \lim {\...
0
votes
0answers
11 views

Equality of second-derivative-like limits of quotients [on hold]

Do there exist two functions $f$ and $g$ such that, for some value of $x$, at least one of the two limits $$\lim_{h \to 0}\frac{\frac{f\left(x+2h\right)-2f\left(x+h\right)+f\left(x\right)}{g\left(x+h\...
-1
votes
1answer
36 views

Find limits : $\lim_{n\to 0^+}\frac{\ln(n(\zeta(n+1)+\Gamma(n)+s)-1)}{n}$ , $s≥0$ [on hold]

How I can find this limits : $$\lim_{n\to 0^+}\frac{\ln(n(\zeta(n+1)+\Gamma(n)+s)-1)}{n}$$ ,$s>0$ My try : $\lim_{n\to 0^+}\frac{\ln(n(\zeta(n+1)+\Gamma(n)+s)-1)}{n}$ $=\lim_{n\to 0^+}\frac{\...
1
vote
2answers
89 views

Finding $\lim\limits_{a\to 1}\int_{0}^{a}x\ln(1-x)dx$

Calculate $$\lim_{a\to 1}\int_{0}^{a}x\ln(1-x)dx, a\in (0,1)$$ I calculate the integral but when I calculate the limit I get $\ln(0)$ and the limit should be $-\frac{3}{4}$. How to approach the ...
0
votes
0answers
16 views

Finding a limit involving F(x) when certain conditions are given

I thought to determine the function first but5 since only one information is given and according to that f(x) has one root alpha and at that point, the derivative has to be zero. So I tried to assume ...
-3
votes
1answer
36 views

Prove that for each integer $m$, $ \lim_{u\to \infty} \frac{u^m}{e^u} = 0 $ [duplicate]

I'm unsure how to show that for each integer $m$, $ \lim_{u\to \infty} \frac{u^m}{e^u} = 0 $. Looking at the solutions it starts with $e^u$ $>$ $\frac{u^{m+1}}{(m+1)!}$ but not sure how this is ...
1
vote
1answer
24 views

Sequence Limit Reciprocal Law Proof

I am aware that this has a duplicate but I am trying to prove it differently than others. Proposition: If $\lim_{n\to\infty} a_n = A \neq 0,$ then $\lim_{n\to\infty} \frac{1}{a_n} = \frac{1}{A}$ ...
0
votes
3answers
52 views

For what values of $x$ is the $\sum_{n=1}^\infty \frac{(7x)^n}{n!}$ convergent?

I was wondering for which values of $x$ the following series converges: $$\sum_{n=1}^\infty \frac{(7x)^n}{n!}.$$ I applied the ratio test to get $$\lim_{n\to \infty} \frac{7x}{n+1}$$ So then $7|x|&...
0
votes
2answers
34 views

Is there a way to state $\limsup_{n\to\infty}{x_n}=x$ using “\forall”

Denote $\{x_n\}$ as a sequence of real numbers, and there exist a $x\in\mathbb{R}$ such that $\limsup_{n\to\infty}{x_n}=x$. We can write "$\lim_{n\to\infty}x_n=x$" as "$\forall \epsilon>0,\exists N\...
1
vote
1answer
33 views

Limit question concerning a triangle and its circumcircle

$ABC$ is an isosceles triangle ($|AB|=|BC|$). Let $s$ be the length of the altitude from vertex $B$ to side $AC$, and let $m=|AC|$. Given that the radius of the circumcircle of $ABC$ is $2\,\text{cm}$,...
0
votes
0answers
40 views

What is the definition of limit in set theory?

I have read at the popular maths site "infinity plus one" that besides the famous "epsilon delta " definition there is another definition which is not always equivalent to the definition of Bolzano ...
-1
votes
1answer
19 views

How do I show that a sequence is not a cesaro summable sequence. [on hold]

How do I show that sequence $\{a_n\}$ where $a_n = (n+1)/2$ if $n$ is odd and $a_n=−(n/2)$ if $n$ is even is not a ($C$,$1$) Cesaro summable sequence (through definition of Cesaro sequences)?
2
votes
0answers
20 views

Limit with factorial and summation [duplicate]

Finding $$\lim_{n\rightarrow \infty}\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}$$ Try: $$\sum^{n}_{k=0}\frac{\binom{n}{k}}{n^k(k+3)}=\frac{n^3}{(n+1)(n+2)(n+3)}\sum^{n}_{k=0}(k+1)(k+2)\frac{\binom{n+...
-1
votes
3answers
28 views

Evaluate the limit using power series without L'Hospital's Rule [on hold]

I'm a bit stumped on this one. Show that $\lim_{x\to0} \frac{e^x -1}{\sin(x)} = 1$ using power series. The instructions are not to use L'Hospital's Rule. I cannot find a way to do this without L'...
0
votes
1answer
44 views

Given a function $f(x)$ that verifies the following conditions

I have a function that verifies the following conditions: $f(x)$ is even $f(5)=6$ $f(x)$ belongs to $[0,5)$ for $x \in [-2,-1]$ It increases in $(- \infty ,-6)$ $\lim_{x\to 6+ } = + \infty$ The ...
1
vote
1answer
41 views

Proving a specific statement involving limits related to a function with some given properties.

Let $n\in\Bbb N$. Let a function $f(x)$ be bounded in every interval $(x_0, x_1)$. The domain of $f(x)$ is $x\in (x_0, +\infty)$. Prove that if the following limit exists and is either finite or ...
1
vote
0answers
33 views

Evaluating limit of a function with integral involved.

We need to evaluate the following limit: $$\lim\limits_{x\to 0} \frac{1}{x} \int_0^x \sin^2\left(\frac{1}{y}\right)\,\mathrm dy.$$ I am finding hard to solve this integral as I cannot see a clear ...
0
votes
2answers
36 views

Proving $\lim_{x\to\infty} e^{\frac{1}{x}} = 1$ using $\epsilon$ (Cauchy definition for limits)

I'm trying to prove that $\lim_{x\to\infty} e^{\frac{1}{x}} = 1$ using the $\epsilon$ definition of diverging to $\infty$. My attempt: (Sorry in advance for English mistakes). For a given $\epsilon ...
1
vote
1answer
37 views

Does $\lim_{x\to x_0}f(x) = a$ and $\lim_{t\to t_0}g(t) = x_0$ imply $\lim_{t\to t_0}f(g(t)) = a$?

Let: $$ \begin{align*} &\lim_{x\to x_0}f(x) = a \tag1 \\ &\lim_{t\to t_0}g(t) = x_0 \tag2 \end{align*} $$ Does $(1)$ and $(2)$ imply the following: $$ \lim_{t\to t_0}f(g(t)) = a\tag 3 $$ ...
0
votes
1answer
17 views

Shouldn't we check for conditionally convergent in ratio test done to see the intervals of convergence in power series?

(By A(n) I mean the power series)I understood that we use absolute value of A(n+1)/A(n) in ratio test because A(n) isn't neccessarily a positive value. We know when there is a limit of absolute value ...
6
votes
2answers
97 views

$\frac1n\sum _{k=1}^na_k\to0$ if and only if $\frac1n\sum _{k=1}^na^2_k\to0$ [duplicate]

If $(a_n)$ is a sequence in $(0,1)$, show that $\frac1n\sum _{k=1}^na_k\to0$ if and only if $\frac1n\sum _{k=1}^na^2_k\to0$ My try: $\implies$: Since $a_k\in (0,1)$, we have $0\le\frac1n\sum _{k=1}^...
0
votes
0answers
26 views

Hi guys, I need to learn Limit and Sequences which book do you recommend to read? [on hold]

I need to learn Limit and Sequences which book do you recommend to read? Where might I download "Sequences, Combinations, Limits by S. I. Gelfand, et al. (1969/2002)"? I tried to find but couldnt find
0
votes
2answers
27 views

Given that $f(x)=(2x+1)^3$, find $\int (\lim_{h \to 0} \frac{f(x+h)-f(x)}{8h})\,dx$

I thought this was as simple as: $$ \int \left (\lim_{h \to 0} \frac{f(x+h)-f(x)}{8h}\right)\,dx = \frac{1}{8}\int f'(x)\, dx=\frac{f(x)}{8} + C $$ But the answer is supposed to be: $$ \left (\frac{...
2
votes
0answers
30 views

If $f(x,y)=9-x^2-y^2$ if $x^2+y^2\leq9$ and $f(x,y)=0$ if $x^2+y^2>9$ study what happens at $(3,0)$

If$$f(x,y)=\begin{cases}9-x^2-y^2&\text{if }x^2+y^2\leq9\\0&\text{if }x^2+y^2>9\end{cases}$$study the continuity and existence of partial derivative with respect to $y$ at point $(3,0)$. ...
7
votes
1answer
96 views

Intriguing Limit

Prove that: $$L=\lim_{n\to\infty} \frac {\sqrt 2 n^{\left(n-\frac 12\right)}}{n!}\left(\frac {(2\sqrt[n] {n} -1)^n}{n^2}\right)^{ \frac {n\left(n-\frac 12\right)}{\ln^2 n}}=\sqrt {\frac {e}{\pi}}$$...
1
vote
3answers
31 views

Having trouble while trying to prove the differentiabilty of $x^2\sin{\left(\frac 1x\right)}$

Let a function be defined as: $ f(x)=x^2\sin{\left(\frac 1x\right)}$ for $x \neq 0$ and $ f(x)=0$ for $x=0$ I'm trying to prove that f is differentiable at 0 using the definition of derivative. ...
2
votes
2answers
34 views

Can dominated convergence justify commuting two limits?

I am evaluating an expression of the form: $$\lim_{a\to 0^+}\,\,\lim_{b\to 0^+}\,\,\sum_{n=1}^\infty\int_{-\infty}^{+\infty}f_n(x;a,b)\,dx.$$ Suppose I can find dominating functions $F_n(x)$ such ...
0
votes
0answers
19 views

Let $\lim_{t\to t_0}\phi(t) = a$. Prove that $f(x) = \mathcal{o}(g(x)) \implies f(\phi(t)) = \mathcal{o}(g(\phi(t)))$

Let: $$ \lim_{t\to t_0}\phi(t) = a $$ where $\phi(t)\ne a$ and $t\ne t_0$ in the neighbourhood of $t_0$. Prove that: $$ \begin{align*} f(x) \stackrel{x\to x_0}{=} \mathcal{o}(g(x)) &\implies ...
2
votes
2answers
56 views

Find $x$ where $f(e^{(x+1)})=x-\ln(x)$ approaches one.

Given that $$ x \in [1,\infty) \quad f(e^{(x+1)})=x-\ln(x) $$ and $$ \lim_{x \to a} f(x)=1 $$ Find $a$. I got to the point: $$ \ln(a)-\ln(\ln(a)-1)=2 $$ But from there on I could not get to $a=e^...
1
vote
1answer
32 views

Solve the following limit as $\lim_{x \to 0}$ [duplicate]

$$\lim_{x \to 0} \frac{x\sin(\sin x) - \sin^2 x}{x^6}$$ **My Attempt: ** I started with L'Hopital's rule. But it quickly became messy. So, I did not continue. I tried to write the Taylor series of ...
3
votes
1answer
67 views

How to solve the given limit?

$$\lim_{n \rightarrow \infty}n^2 \int_{0}^{1} \frac{1}{(1+x^2)^n}dx$$ How to solve this,what should be our first approach?
-2
votes
4answers
63 views

What's “limit doesn't exist” multiplied by limit that equals zero?

OK, we're having a strong discussion just a day before our Calculus exam. The problem's next: To check if this function is continuous: $$\frac{y^2\,\sin x}{x^2 + y^2}$$ at (0,0). We get DNE $\cdot 0$, ...
4
votes
2answers
56 views

$\lim_{\epsilon\to0}\frac{\cos(\epsilon-n\frac{\pi}{2})}{\epsilon^n}$

We were doing generalized integrals in class and this integral came out. I tried using integration by parts and got something repeating. We're gonna let $\epsilon \rightarrow 0$ and $x\rightarrow\...
4
votes
2answers
212 views

Can we cancel the equality mark here?

Problem Let $f(x)$ satisfy that $f(1)=1$ and $f'(x)=\dfrac{1}{x^2+f^2(x)}$. Prove that $\lim\limits_{x \to +\infty}f(x)$ exists and is less than $1+\dfrac{\pi}{4}.$ Proof Since $f'(x)=\dfrac{1}{x^2+...
1
vote
3answers
66 views

Limit of $\sin(xyz)/xyz$

What is the limit of $\dfrac{\sin(xyz)}{xyz}$ when all $x,y,z$ go to zero? P.S. I think the answer is $1,$ since the limit of $\sin(m)/m$ is zero as $m$ goes to $0$.
0
votes
4answers
39 views

Evaluating limits in fractions

When you want to find the limit of a fraction e.g. $\frac{1-x}{1-x^3}$ as $x$ tends to $1$. Why can you not just plug in x into the numerator and denominator? Why do you have to make all the $x$ ...
2
votes
4answers
39 views

How to solve non-fractional limits like $\lim_{x \to \infty} x^{(\ln5) \div (1+\ln x)}$?

I have the limit $\lim_{x \to \infty} x^{(\ln5) \div (1+\ln x)}$. I am trying to figure out how to solve this, but I only know how to handle limits when they can be made into fractions. Is there some ...
1
vote
1answer
55 views

Prove that this sequence of continued fractions $\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}},\dots$ tends to $1$.

The Problem: I'll write up a couple more terms: $$\frac{2}{1}, \frac{6}{5+\frac{4}{3}}, \frac{12}{11+\frac{10}{9+\frac{8}{7}}}, \frac{20}{19+\frac{18}{17+\frac{16}{15+\frac{14}{13}}}}, \frac{30}{29+\...
5
votes
2answers
83 views

Limit of $\left\lfloor x \left\lfloor \frac1x \right\rfloor \right\rfloor$, as $x$ goes to zero

Find $\lim\limits_{x\to 0^+}\left\lfloor x \left\lfloor \frac1x \right\rfloor \right\rfloor$ and $\lim\limits_{x\to 0^-}\left\lfloor x \left\lfloor \frac1x \right\rfloor \right\rfloor$ ? See the ...
2
votes
2answers
94 views

Evaluate $\lim_{n\rightarrow \infty }\frac{1}{n}\int_{1}^{n}\frac{x-1}{x+1}dx$

$$\lim_{n\rightarrow \infty }\frac{1}{n}\int_{1}^{n}\frac{x-1}{x+1}dx$$ My approach is not correct, I think. I took $f(x)=(x-1)/(x+1)$ which is continuous so there is a $F(x)$ a primitive of f(x) ...
2
votes
0answers
38 views

Proving a function is stable if and only if $|f'(x)| \geq c$.

A function $f : \mathbb{R} \rightarrow \mathbb{R}$ is stable provided that $$|f(x) - f(y)| \geq c|x - y|$$ for all $(x, y) \in \mathbb{R}^{2}$, where $c > 0$ is called the stability ...
4
votes
1answer
82 views

Prove that $\lim_{x\to\infty}\sum_{n=1}^{\infty}\frac x{n^2+x^2}$ exists and is positive

Show That $$\sum_{n=1}^\infty{1\over x ^2+n^2} \sim \frac1x$$ as $x\to \infty.$ It is enough to show that $\lim_{x\to\infty}\sum_{n=1}^{\infty}\frac x{n^2+x^2}$ exists and is positive $$=\lim_{x\to\...
4
votes
1answer
37 views

Proving the well-definedness of $df$: How to place the limit inside the argument of $\psi \circ f \circ \phi^{-1}$?

In the book of Chillingworth, the author defines the tangent space of a point $p$ in the smooth manifold $M$ as the set of all conjugacy classes of smooth paths with $\alpha (o) = p$ s.t $\alpha \sim \...
1
vote
0answers
44 views

How to calculate the following limit in a clever way?

$\underset{x\to 0}{\text{lim}}\frac{x^{(\sin x)^x}-(\sin x)^{x^{\sin x}}}{x^3}=\frac{1}{6}$, the limit is easy to get results.but how to rigorously prove it without using Taylor formula ? At ...
-6
votes
1answer
42 views

Limit of $(1-(\cos(a/n))^{2n})/\tan(a/n)$ [on hold]

Anyone that can help me to prove that $$\lim_{n\to\infty} \frac{1-(\cos(a/n))^{2n}}{\tan(a/n)}=a\pi$$ (Preferably without l'Hospital's Rule.) TIA
1
vote
1answer
46 views

Proving that $\lim_{(x,y)\to(0,2)} |y|^x(x+1)^y = 1$ using the limit definition

I started with this strategy: $$||y|^x(x+1)^y - 1| = \bigg||y|^x \bigg((x+1)^y -\frac{1}{|y|^{x-1}} + \frac{1}{|y|^x} \bigg) + |y| - 2 \bigg| \leq \bigg||y|^x \bigg((x+1)^y -\frac{1}{|y|^{x-1}} + \...
0
votes
2answers
35 views

A homework question about partial limits

I could really use some help figuring out this question. The question: ${a_n}$ is a series so that $\lim_{n\to\infty} (a_{n+1} - a_n) = 0$. Prove that its group of partial limits is the closed ...
2
votes
3answers
66 views

Finding $\lim_{x \to \infty} \sqrt{x} c^x$ for $0<c<1$

Is there a short way to prove that $\lim_{x \to \infty} \sqrt{x} c^x = 0$ for $0<c<1$? I tried using L'Hospital's rule and a few substitutions, and even if I was getting somewhere the proof was ...