Questions on the evaluation and properties of limits.

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Heaviside function & Integral Limits

When considering integration, how does one use the Heaviside function in order to alter the limits of integration. For example If i have $$ \int_a^b f(x) dx $$ But want to change this integral to be ...
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1answer
30 views

What sequence of polynomials is equal to $2^n$ for integers $1$ to $k$?

I am trying to prove to someone that no matter how many terms you have of a sequence you can never be 100% sure of the underlying formula. Consider this sequence: $$2^n=1,2,4,8,16,...$$ But just given ...
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0answers
66 views

Is it possible to find $\lim_{n\rightarrow\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}} $ without using integral or combinatory logic? [on hold]

$$\lim_{n\rightarrow\infty}\frac{1^n+2^n+3^n+\ldots+n^n}{n^{n+1}} $$ I apologize for asking the same question again, but I wanted to ask something. Is there a possibility that this problem could be ...
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2answers
61 views

What is the largest function whose integral still converges?

Let C be the set of all functions $f(x)$ whose integral converges, i.e. for some constant $x_0$: $$\int_{x_0}^\infty f(x) dx < \infty$$ While playing with integrals in Wolfram Alpha, I noticed ...
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0answers
15 views

Limit and Continuity dependency [on hold]

Whether someone knows any interesting example of dependency between limit and continuity of function? Thank you for any proposition
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35 views

Are these two limits equal to each other?

I'm curious about whether these two limits are the same (well I know they are equal since Wolfram Alpha confirms it, but I want to know whether the reasoning is justified): $$ \lim_{x\rightarrow ...
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1answer
28 views

Some conditions on $\tilde f(x,y)=\begin{cases}\displaystyle g(x,y) & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$

The following function $$f(x,y)=\begin{cases}\displaystyle\frac{x^2 y^2}{x^2+y^2} & \text{if }(x,y)\not=(0,0), \\ 0 & \text{if } (x,y)=(0,0).\end{cases}$$ is differentiable in the origin and ...
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1answer
21 views

Measurability of sequence of functions

Let $(f_n)_{n \in \Bbb N}$ be a sequence of measurable functions on a measure space $(X, M, \mu)$. Prove that the set $\{x \in X \; | \; \lim_n f_n(x) \text{ exists} \text{in } [-\infty, ...
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1answer
29 views

Trying to understand Spivak's answer for limit proof (Chapter 5 problem 3v)

Prove the limit l for the function at a: $$f(x)= x^4 + \frac1x, a =1$$ I have successfully found a $\delta$ in terms of $\epsilon$, and here is how I did it: Since we can see the limit is 2 at a = ...
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1answer
58 views

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? [duplicate]

Does the improper integral $\int_{0}^{\infty}\sin(x^2)\;\mathrm dx$ converge? So if it converges then $\lim_{b \to\infty}\int_{0}^{b}\sin(x^2)\;\mathrm dx$ exists and our integral converges to this ...
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1answer
26 views

Integration: Finding area, volume and arc length

I am new to integration, so please do not mark this question as "not enough research done" Here is the question (please open image in new tab to see it clearly) - I am getting stuck with the ...
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1answer
24 views

Prove or disprove this relation between one root of the quadratic and the cubic equation of a certain form, and linear recurrences.

It is well known that the n-anacci (higher degree Fibonacci, that is Tribonacci and so on) numbers can be computed in closed form from roots of polynomials in the way Eric Weisstein at Mathworld ...
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3answers
29 views

How to prove that those limits are equal?

let $\lim a_n = a$ and $\lim b_n = b$. We define two groups:- Group A:- Includes all the elements that imply $a_n>b_n$.All the elements in $a_n$ that are bigger than the the elements in $b_n$. ...
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0answers
38 views

Integration help needed with two problems

Here are 2 problems that are confusing me to a great extent - I have no issues with integration, but I'm unable to figure out the limits and the equation to be integrated in these problems. The ...
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1answer
21 views

General technique of proving limit exists as x approaches infinity

I'm looking for a way of proving that $f(x)$ will have some limit (without specifying what it is) as $x \to \infty$. To make this more concrete, I'm asked to (1) prove the limit exists and (2) find ...
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1answer
53 views

How can you find the integral of $\frac{cos(2t)}{2t^2}$ between 1 and infinity?

How can you find the integral of $\frac{\cos(2t)}{2t^2}$ between 1 and infinity? $$ I = \int\limits_1^\infty \frac{\cos(2t)}{2t^2} dt $$ My problem is that I just simply do not know how to handle ...
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1answer
38 views

Can someone help me with this limits problem

$$\lim_{x\to0^+}\frac{\sin \{\sqrt{x}\}}{\{\sqrt{x}\}}$$ Where $\{x\}$ is the decimal part of $x$. How do I go about solving this problem? Thank you
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1answer
35 views

A Limit of a Geometric Average

I have a problem calculating the following limit: $\lim\limits_{n \to \infty}{ (1-2/3)^{3/n}*(1-2/4)^{4/n}...(1-2/(n+2))^\frac{n+2}{n}}$ I thought this is a geometric average of the first n items of ...
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4answers
52 views

Prove that $\lim_{n\to \infty} n\cdot r^n=0$ where $(0\leq r <1)$ without using ratio test

$\lim_{n\to \infty} n\cdot r^n=0$, where $0\leq r <1$, can be obtained by vanishing condition (considering $\sum^{\infty}_{n=1}n\cdot r^n$, which converges, using ratio test). Is there a direct ...
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1answer
22 views

limits as $x\rightarrow\pm\infty$ of indeterminate forms $\frac{a^x+b^x}{c^x+d^x}$, where $a,b,c,d\in\mathbb{R}$

Good day sirs would you kindly help me to find the limit of $\frac{a^x+b^x}{c^x+d^x}$ as $x\rightarrow\pm\infty$, where $a$,$b$,$c$ and $d$ are real numbers? I already know how to use the L' ...
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1answer
49 views

If $\limsup_n\sqrt[n]{a_n}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{(n+1)a_{n+1}}=?$

If $\limsup_n\sqrt[n]{|a_n|}=\frac{1}{r}$, then $\limsup_n\sqrt[n]{|(n+1)a_{n+1}|}=?$ Can I separate the product of the second limit as ...
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0answers
15 views

What is the limit of this divergent infinite product multiplied by an exponential?

What is... $$\lim_{\omega \to \infty} \left( {1 \over {c^{\omega}}} \cdot \prod_{N=1}^{\omega} (1+e^{b \cdot c^{-N}}) \right)$$ My attempt: I have absolutely no clue except for the case of $c=2$ ...
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1answer
32 views

Proving the equality of sequences' limit [duplicate]

Prove that if $a_n$ is convergent, then $M_n:=\frac{1}{n}\sum_{1}^{n} a_n$ satisfies $\lim M_n=\lim a_n$. (sorry for English)
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1answer
52 views

How to check that the limit $\lim_{ x \to \infty }f(x)$ exists?

Let $f:[1, \infty ) \rightarrow R$ and $C^1([1, \infty ))$ such that $f'(x)=\frac{1}{x^2 +\sin^2x+f(x)}$ Prove $\lim_{x \to \infty }f'(x)=0$. How to check that the limit $\lim_{ x \to \infty }f(x)$ ...
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1answer
21 views

Iterated limits of $\frac{x-y}{x^3-y}$

Why it the following limits look like this: $$\lim_{x\rightarrow -1} \frac{x-y}{x^3-y}=1$$ but suprisingly $$\lim_{y\rightarrow -1} \frac{x-y}{x^3-y}=\frac{1}{1-x+x^2}$$I thought that after ...
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1answer
56 views

Verification of solution of a contest problem with a limit of nested radicals

They gave me 0 points for this problem. I think it's unfair. What do you think of this proof, is it correct? $\lim\limits_{n\to\infty} \underset{2n\text{ roots ...
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1answer
36 views

Prove limit exists if and only if left and right limits exist and are equal

Prove $\lim_{x\to a}f(x)=L \iff \lim_{x\to a^+}f(x)=L=\lim_{x\to a^-}f(x)$ I have no problem with the $(\Leftarrow)$ direction but I can't do it for the other direction. Proofs for both directions ...
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0answers
24 views

uniform limits and asymptotic equivalence

I am told that $\frac{f(\lambda r)}{f(r)}$ tends to 1 uniformly in $\lambda$. I also know that $x(t)$ is asymptotically equivalent to $ct$, so $x(t)\sim ct$. How can I show that $\frac{f(x(\lambda ...
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2answers
48 views

Finding the limit of $\log(1+ax)/\log(1+x)$ [on hold]

I want to find the following limit: $$\lim_{x\to\infty}\frac{\log(1+ax)}{\log(1+x)}$$
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1answer
51 views

Limit comparison test

Since the series $\sum_{n=1}^\infty a_n$ is convergent, so $\lim_{n\to\infty} a_n=0.$ Consider $$\lim_{n\to\infty} {\left(\dfrac{{a_n}^{1/2}}{n}\right)\over{\left(\dfrac{1}{n^2}\right)}} $$ which is ...
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3answers
43 views

Finding the Limit of the Ratio of a Recursive Sequence's Terms

Let {$f_n$} be defined recursively as $f_1 = f_2 = f_3 = 1$ and $f_n = f_{n-1} + f_{n-3}$ for all $n \gt 3$. Also, define {$a_n$} as the ratio of the terms of {$f_n$}. That is, $a_n = ...
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2answers
47 views

Does this limit exist finitely?

Find the value of the following (if it exists): $$\lim_{n\to\infty}n\sin(2\pi en!)$$ Does it exist? I think that it doesn't exist but I can't prove it. Please help me.
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1answer
42 views

A debate over the limit of $ \frac{f(x + a h) - f(x + b h)}{h} $ as $ h $ approaches $ 0 $.

This may seem like an easy question, but a few of us are having a debate over it. We are looking at the following limit below, where $ f $ is a real-valued function on an open subset $ U $ of $ ...
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2answers
34 views

Find if the limit of $\lim_{ (x,y) \to (0,0)} \frac{xy}{\sqrt{x^2+y^2}}$

If we approach $f(x,y)=\frac{xy}{\sqrt{x^2+y^2}}$ on $y=mx^n, n \in R, n>2$ Then we get $\lim_{ (x,y) \to (0,0)} \frac{x(mx^n)}{\sqrt{x^2+m^2x^{2n}}}$ $\lim_{ (x,y) \to (0,0)} ...
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3answers
56 views

How to find if $\lim_{n\to \infty} {n \over 3^n}$ is convergent or divergent

$$\lim_{n\to \infty} {n \over 3^n}$$ I think this involves the comparison test? My thought was comparing it to $1 \over 3^n$
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2answers
89 views

the lim of sum of sequence

I have to calculate the following: $\lim_{n \to \infty} (\frac{n}{n^2+1^2}+\frac{n}{n^2+2^2} + \cdots + \frac{n}{n^2+n^2} )$ I managed to understand that it is $\lim_{n \to \infty} ...
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4answers
60 views

Calculate $\lim_{x\, \to \,\infty} \bigg(\frac{\sqrt[x]{1} + \sqrt[x]{2}}{2}\bigg)^x$

I am just tryin to solve the limit: $$\lim_{x\, \to \,\infty} \bigg(\frac{\sqrt[x]{1} + \sqrt[x]{2}}{2}\bigg)^x$$ (hope this isn't a duplicate, it is quite complicated to find special eq's via the ...
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1answer
18 views

Proving the result of the limit

How can I prove the this limit is 0 ? My experience says that the limit is zero because of the velocity that the denominator goes to zero, but I can't prove it with calculus. $$\lim_{x\to ...
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0answers
21 views

Oscillations of a feedback interconnection

I have a feedback interconnection described via the following transfer function $G(s)=\frac{1}{s^3+5s^2+6s+1}$ and the nonlinearity $\psi(e)=\text{sgn}(e)$. I have used the describing function method ...
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1answer
66 views

Evaluating $\lim_{x\to 0} x^x$

$$ x ^{x } = e^{\ln x^x } $$ $$ \lim_{x\rightarrow 0} x^x = \;? $$ I need to find the limit of x to the power of x as x approaches to 0 using l'Hopital's rule. From previous part there is a hint that ...
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1answer
37 views

Two functions equal in some point

I have two continuous functions $f,g$, $f(0) \lt g(0), f(1) \gt g(1)$. How do I prove without using "advanced" theorem (using only definitions of limit, continious functions and sup/inf definitions), ...
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0answers
15 views

Solve a high order polynomial equation in $x$ in the limit $n\rightarrow\infty$

A bit of background. I did a high order WKB theory to calculate the eigenvalues of a potential. The eigenvalues, $E$, are, of course, real since they correspond to a physical problem. My final answer ...
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2answers
27 views

Do equal limits from orthogonal directions imply the limit of the function?

If a function approaches the same limit from orthogonal directions, does the limit then exist? For example, take $f:\mathbb{C}\rightarrow\mathbb{C}$ where $f(z)=\frac{\sin{z}}{z}$. Does the fact that ...
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0answers
25 views

Geometric distribution converges to exponential distribution

For $n\in \mathbb{N}$ let $X_n$ be geometric with parameter $p_n \in (0,1)$, that means $\mathbf{P}[X_n = k] = p_n(1-p_n)^k$, $k\in\mathbb{N}_0$. How must the sequence $(p_n)_{n\in\mathbb{N}}$ ...
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3answers
41 views

If the limit of the sequence exists, find it. If not, prove that the limit does not exist. [on hold]

Consider the following sequence: $\{[\sqrt{n}][\sqrt{n + 1}-\sqrt{n}]\}$ for $ n \geq 1$. If the limit exists, find it and prove that the limit is indeed your choice. If not, prove that the limit ...
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1answer
31 views

Are there general guidelines to make “assumptions” when proving limits?

I am studying the definition of the limit using Paul's Online Notes When proving the following limit (Example 3) $\lim\limits_{x \to 2} x^2+x-11 = 9$ At one point he assumes: $|x+5| < K$ He ...
3
votes
3answers
76 views

Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$

Suppose that $f(a)=0$. Prove that $g(x):=|f(x)|$ is differentiable iff $f'(a)=0$ Not sure how to go about this at all. The limit definition that I am working with is $$ g'(a)=\lim_{x \rightarrow ...
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1answer
23 views

Proof of $f'(x) = \frac{1}{n} \cdot x^{\left(\frac{1}{n}\right) - 1}$ when $f(x) = x^{\frac{1}{n}}$ by limit definition [on hold]

how can I prove $$f'(x) = \frac{1}{n} \cdot x^{\left(\frac{1}{n}\right) - 1}$$ when $$f(x) = x^{\frac{1}{n}}$$ by limit definition? Thank you.
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3answers
81 views

Find $\lim\limits_{n\rightarrow\infty}(n!)^{1/n^2}$ [on hold]

I need some help with this limit $\lim\limits_{n\rightarrow\infty}(n!)^{1/n^2}$, , I was thinking how to do, but I have no idea how to start.
0
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2answers
58 views

$\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$

I was trying to do this limit $$\lim\limits_{n\rightarrow\infty} \sqrt{n}(\sqrt{n+1}-\sqrt{n})$$ and I cant, I will be pleased if someone give me a hint to do it.