# Tagged Questions

Questions on the evaluation of limits.

2answers
33 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 $\lim_{x\to x_0} f(x)=f(x_0)$. If $f(x)$ is ...
5answers
96 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 ...
2answers
109 views

### High school contest question

Some work on it reveals the possibility of using gamma function. Is there any easy way to compute it? $$\lim_{n\to\infty}\left(\frac{1}{n!} \int_0^e \log^n x \ dx\right)^n$$
1answer
30 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 ...
5answers
88 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.
1answer
19 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 ...
3answers
62 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? ...
2answers
60 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 ...
2answers
53 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 ...
8answers
130 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} ...
4answers
84 views

### How to evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?

How can I evaluate $\lim_{x \to \infty}\left(\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}\right)$?
1answer
59 views

### Why do these trig functions “overpower” each other?

For example, $\sin(x)\cos(x)$ can be written as $\sin(2x)/2$, the limit as $x$ approaches $0$ of $\sin(x)\cos(x)$ is $0$, and the limit as x approaches $\pi/2$ is $0$. I don't see a reason why sine ...
1answer
45 views

### limit of an exponential function

I was trying to understand how we can approximate exp. One example is: $$\exp(t) = \sum_{i=0}^\infty t^i/i!$$ however, why is the next true: \lim_{x\to \infty}\exp \left ({\frac{t^2}{2!} ...
3answers
84 views

### What does this mean: there exist an integer N such that $n\ge N$?

I'm reading Rudin's, Principles of Mathematical Analysis, and I keep tripping over this phrase. Usually the phrase by that it implies a some equation with n being the index, subscript, of a point. My ...
1answer
17 views

### Decreasing from the horizontal asymptote

The function $f(x) = x^2/(x^2 - x -2)$ has the following graph. It has a horizontal asymptote $y=1$. For $x$ less than $-4$, the function is decreasing and its graph is under the asymptote. How is ...
3answers
30 views

### Finding the limit of a particular function

Can not understand the following limit in a past paper. $$\dfrac{1}{[n(e^{\frac{1}{n}}-1)]}\rightarrow 1$$
7answers
124 views

### How to prove that $1/n!$ is less than $1/n^2$?

I want to prove $$\sum_{n=0}^{\infty} \frac{1}{n!}$$ is a converging series. So I want to compare it with $\sum_{n=0}^{\infty} \frac{1}{n^2}$. I want to do direct comparison test. How to prove ...
2answers
49 views

### How to find the limit for the quotient of the least number $K_n$ such that the partial sum of the harmonic series $\geq n$

Let $$S_n=1+1/2+\cdots+1/n.$$ Denote by $K_n$ the least subscript $k$ such that $S_k\geq n$. Find the limit $$\lim_{n\to\infty}\frac{K_{n+1}}{K_n}\quad ?$$
3answers
46 views

### Evaluating a limit with variable in the exponent

For $$\lim_{x \to \infty} \left(1- \frac{2}{x}\right)^{\dfrac{x}{2}}$$ I have to use the L'Hospital"s rule, right? So I get: $$\lim_{x \to \infty}\frac{x}{2} \log\left(1- \frac{2}{x}\right)$$ And ...
1answer
49 views

### How to place a limit that it's inside the integral, outside.

I did this: $$\int_{1}^t x^{-1}dx=\int_{1}^t\lim_{n\rightarrow -1}{x^n}dx =\lim_{n\rightarrow -1}\int_{1}^t{x^n}dx$$ just to have a way to approximate $\ln t$. \ln{t}=\lim_{h\rightarrow ...
0answers
58 views

### Find the limit $\lim_{n\to\infty}\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}}$ [duplicate]

Find the limit $$\lim_{n\to\infty}\sqrt{1+\sqrt{2+\cdots+\sqrt{n}}}$$. Remark:there are n times square root within $n$.
5answers
42 views

### How to find the limit of this function

We have the function $$\dfrac{\sqrt{n^4 + 100}}{4n}$$ I think the best method is by dividing by $n$, but I have no idea what that yields, mainly because of the square root.
4answers
117 views

### What is $\lim_{n \rightarrow \infty}\sum\limits_{k = 1}^n\frac{k}{n^2}$?

We have $$\dfrac{1+2+3+...+ \space n}{n^2}$$ What is the limit of this function as $n \rightarrow \infty$? My idea: \dfrac{1+2+3+...+ \space n}{n^2} = \dfrac{1}{n^2} + \dfrac{2}{n^2} + ... + ...
1answer
29 views

### Find the limit as n approaches infinite

We have the following function: $$U_n = \sin \dfrac{1}{3} n \pi$$ What is the limit of this function as n approaches infinity? I first tried to use my calculator as help, for n I chose some ...
3answers
34 views

### Sequence of the ratio of two successive terms of a sequence

If $(a_n)_{n\in N}$ is a strictly decreasing sequence of real number converging to $0$ and s.t. $\forall n\in N$, $0<a_n<1$, does the following limit: $$\lim_{n}\frac{a_{n+1}}{a_n}$$ exists? ...
2answers
42 views

### How to calculate $\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}}$ when $x>1$?

Numerically, it looks that the limit is $$\lim_{x\to 1}\int_{0}^{1}\frac{dy}{\sqrt{1-y^{2}}}\frac{y^{3/2}}{\sqrt{x - y}} = \frac{1}{\sqrt{2}}\log(1 - x) + cte$$, but I have not been able to ...
2answers
31 views

### limit of convex increasing funtion

If $f$ is strictly increasing and strictly convex (or $f'>0$ & $f''>0$), then $$\lim_{x\rightarrow∞}{f(x)}=∞$$ Is this statement true? If this statement is true, how can I prove?
2answers
92 views

### Which is the better approximation to $e$?

Let $a_n = (1+1/n)^n$ and $b_n = (1+1/n)^{n+1}$. Both $a_n \to e$ and $b_n \to e$, and $a_n < e < b_n$. A better approximation to $e$ is known to be $c_n = (1+1/n)^{n+1/2} = \sqrt{a_n b_n}$, ...
2answers
43 views

### multivariable limit question

Is this an acceptable solution? $$\lim_{(x,y)\rightarrow(0,0)}\frac{\sin(2(x^2+y^2))}{x^2+y^2}$$ $$t=x^2+y^2$$ So $t\rightarrow0$. Now I change the limit to: ...
3answers
52 views

### Limit as N goes to Infinity

Consider this limit: $$\lim_{n\rightarrow\infty} \left( 1+\frac{1}{n} \right) ^{n^2} = x$$ I thought the way to solve this for $x$ was to reduce it using the fact that as $n \rightarrow \infty$, ...
2answers
110 views

### Without calculating limit directly show that it is equal to zero

$$\lim_{n\rightarrow\infty}\left(\frac{n+1}{n}\right)^{n^2}\frac{1}{3^n}=0$$ I am not really sure what it means by "without calculating limit" and I don't really have ideas how to do it.
3answers
106 views

### How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$?

How to calculate $\lim_{x\to 1^+} \log (x)^{\log(x)}$ ? i know that its "1", but why? How can i calculate this? Thank you very very much =)
1answer
94 views

### What is this limit called? Is it a different kind of derivative?

(first I should notice you this is not something I can look up in a textbook, because I'm learning partial derivatives, alike I do with most Maths, as a hobby. If something below is wrong, blame the ...
1answer
14 views

### Ignoring exponential terms in asymptotic matching of two point boundary value ODE

So I'm not sure how much background I need to give to set up this question. But in my lecture notes I have that $e^{-\eta / \epsilon^{1-\alpha}}$ can be ignored where $\epsilon << 1$ and \$0 ...
2answers
42 views

### Find the limit of $2+\left(-\frac{2}{e}\right)^n$, as $n\to\infty$, if it exsists

I'm absolutely unsure about how to approach this. I've considered changing it to $-2=\left(-\frac{2}{e}\right)^n$ and then using the properties of lograrithms, but $\ln(-2)$ is undefined, as is ...
2answers
36 views

### Is it ever proper to say that the limit of a function equals infinity?

If I calculate a limit and get the value $\infty$, what is the proper way to communicate this? Can I say that the $\lim_{n\to\infty}a_n=\infty$ and therefore the sequence $\{a_n\}$ diverges, or do I ...
3answers
27 views

### Limit as n approaches infinity involving roots

$$\lim_{n\to\infty}\frac{n}{1+2\sqrt{n}}$$ Given my understanding of how to solve these problems, I need to take the highest power of $n$ in the denominator and then divide both the numerator and ...
1answer
51 views

### accumulation point of recursive sequence

Given is a sequence with: $(a_0)=1$, $(a_1=1)$, $a_{n+2}=\frac{1+a_{n+1}}{a_n}$ I now have to show what the accumulation points are: I guess that the sequence is jumping from number to number like ...
4answers
568 views

### Find $\lim\limits_{n \rightarrow \infty}\dfrac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}$

Find$$\lim_{n \rightarrow \infty}\frac{\sin 1+2\sin \frac{1}{2}+\cdots+n\sin \frac{1}{n}}{n}.$$ This is a recent exam question, which I couldn't figure out in the exam. My guess is it doesn't ...
1answer
39 views

### Show that sequence approaches fixed point of a function

Problem Let $f(x)$ be a differentiable function on $\Bbb R$ with $\left|\,f ' (x)\right| \leq r < 1$, where $r$ is constant. Then consider the sequence $\{x_n\}$ such that $x_1 = 0$, \$x_{n+1} = ...
1answer
138 views

### What is the most elementary proof that $\lim_{n \to \infty} (1+1/n)^n$ exists?

Here is my candidate for the most elementary proof that $\lim_{n \to \infty}(1+1/n)^n$ exists. I would be interested in seeing others. $***$ Added after some comments: I prove here by very ...
1answer
38 views

### please,find the limit of $x$

please,find the limit of $x$ when $x=u^{-\frac{1}{\alpha}}$ where ${0}\le u\le{1}$ and $\alpha>0$ i have found the limit of $x$ is also ${0}\le x\le{1}$ when $u=0$ then $x=0$ since ...
1answer
46 views

### How to calculate $\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$?

How can we compute the following limit: $$\lim_{n\to\infty}(1+1/n^2)(1+2/n^2)\cdots(1+n/n^2)$$ Mathematica gives the answer $\sqrt{e}$. However, I do not know to do it.
1answer
61 views

### Solving the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$

Suppose the equation $\displaystyle \frac{e^x}{x}=\int_n^{n+1}f(t)\,dt$ as $f(t)=\frac{e^t}{t}$ and $n\in \mathbb{N} \setminus{0}$. How to prove that: The equation above has a unique solution $U_n$ ...
2answers
18 views

### Limit with variable: non-defined expression

I have a given limit that depends on a variable $a$: $$\lim_{x \rightarrow \infty} \left (\frac{e^{ax}}{1 - ax} \right)$$ I understand cases for $a < 0 \implies \lim = 0$ and \$a > 0 \implies ...
2answers
88 views

### Finding $\lim\limits_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$ for continuous $f:[0,1]\to[0,\infty)$ [duplicate]

Find $$\lim_{n \rightarrow \infty}\left(\int_0^1(f(x))^n\,\mathrm dx\right)^\frac{1}{n}$$if $f:[0,1]\rightarrow(0,\infty)$ is a continuous function. My attempt: Say $f(x)$ has a max. value $M$. ...
5answers
197 views

### Why isn't this limit equal to $0$?

$f(2)=4$, $g(2)=9$, $f'(2)=g'(2)$. $\displaystyle \lim_{x \to 2} \frac{ \sqrt{f(x)}-2} { \sqrt{g(x)}-2}$. Why isn't this limit equal to $0$? Since $f$ and $g$ are differentiable at $x=2$, that ...
1answer
29 views

### Finding distributional limit

How to find $\lim_{\varepsilon\rightarrow 0+}f_{\varepsilon}$ in $D'(R)$, if $f_\varepsilon$ is defined as: $f_\varepsilon(x)=\frac{1}{\varepsilon^3}$ for ...
2answers
41 views

### How to prove something on limit points…

$X = \mathbb R^n$. How to prove that the interior point of a subset of $X$ is also limit point? Don't know where to start... by definition, an interior point is such if it exists an e-neighborhood ...
1answer
32 views

### Are all limits solvable without L'Hospital Rule or Series Expansion

Is it always possible to find the limit of a function without using L'Hospital Rule or Series Expansion For example, $$\lim_{x\to0}\frac{\tan x-x}{x^3}$$ $$\lim_{x\to0}\frac{\sin x-x}{x^3}$$ ...
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