Questions on the evaluation and properties of limits.

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1
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1answer
42 views

$\lim_{x\to{\infty}} (x-\sqrt{x^2+x})$

Given this question $$\lim_{x\to{\infty}} (x-\sqrt{x^2+x})$$ Find the limit Work so far...: $$\lim_{x\to{\infty}} \frac{x^2-x^2-x}{x+\sqrt{x^2+x}}$$ $$\lim_{x\to{\infty}} \frac{-x}{x+\sqrt{x^2+x}} ...
1
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2answers
39 views

Indeterminate form $0^0$ using L'hospitals rule when calculating $\lim_{x\to0^+} x^{\sin(x)}$

Given the question $$\lim_{x\to0^+} x^{\sin(x)}$$ I have deducted so far that this has the indeterminate form $0^0$ so I have taken the natural logarithm of both sides to give me: $$\lim_{x\to0^+} ...
1
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3answers
38 views

Calculate Infinite Limit

I'm trying to calculate the limit and when I get to the last step I plug in infinity for $\frac 8x$ and that divided by -4 I get - infinity for my answer but the book says 0. Where did I go wrong? $$ ...
9
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2answers
74 views

Finding $\lim_{x\to 0} \frac {2\sin x-\sin 2x}{x-\sin x}$ geometrically

While looking at this question, I noticed an interesting geometric interpretation of the limit the OP was trying to evaluate. His limit came to twice the value of the limit $$\lim_{x\to 0}\frac{\sin ...
4
votes
1answer
72 views

Proving a convergence relationship between two sequences

Let $a_{n}$ a sequence of real numbers. Let $\sigma_n= \frac{a_1+a_2+...+a_n}{n}$. Suppose that $\lim_{n\to \infty} \sigma_n=A.$ Prove that $$\lim_{n \to \infty}\frac{1}{\log n} ...
1
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4answers
71 views

Find the limit of $(2\sin x-\sin 2x)/(x-\sin x)$ as $x\to 0$ without L'Hôpital's rule

I wonder how to do this in different way from L'Hôpital's rule: $$\lim_{x\to 0}\frac{2\sin x-\sin 2x}{x-\sin x}.$$ Please help me solve this without using L'Hopital's rule.
0
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1answer
14 views

Convergence and Irrationality of $\frac{H_{(n,-n)}}{(n+1)^n}$ as $n$ approaches infinity

We define $H_{(a,b)}$ as the $a^{th}$ harmonic number of class $b$. In other words, $$H_{(a,b)}=\sum_{k=1}^a \frac{1}{k^b}$$ More information about generalized harmonic numbers can be found here. Let ...
2
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5answers
69 views

How do I prove that $\lim_{(x,y)→(0,0)}\frac{1-\cos(x^2+y^2)}{\sqrt{x^2+y^2}} = 0$

$$\lim_{(x,y)→(0,0)}\frac{1-\cos(x^2+y^2)}{\sqrt{x^2+y^2}} = 0$$ I believe this is correct since I couldn't find a directional limit that won't validate this. From what I know, I have to prove that ...
0
votes
2answers
68 views

Evaluate $\lim_{n\to\infty}\left(\sqrt{\frac{9^n}{n^2}+\frac{3^n}{5n}+2}-\frac{3^n}{n}\right)$

Find, if it exists, the following limit: $\displaystyle\lim_{n\to\infty}\left(\sqrt{\frac{9^n}{n^2}+\frac{3^n}{5n}+2}-\frac{3^n}{n}\right)$.
2
votes
2answers
43 views

Spivak Chapter 5, Lemma 3 - Explanation

I'm having some trouble understanding the 3rd lemma in Chapter 5 (page 101 for the 3rd edition) of Spivak's Calculus, that states: If $\ y_{0} \neq 0 $ and $\ |y-y_{0}| < min(\frac{|y_{0}|}{2}, ...
1
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2answers
38 views

Limits and definition integrals involving logarithms

Let $a \in (0,1)$ and define $$I_n(a)=\int_a^1 (\ln x)^n \, \mathrm{d}x$$ Show that limit as $a\to 0$ we have, $$\lim_{a\to 0}I_n(a)=(-1)^n \cdot n!$$
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votes
4answers
63 views

Find the limit $\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$ [on hold]

I need help finding the following limit. $$\lim_{n \to \infty} \sum_{k=1}^n \frac{1}{k+n}$$
0
votes
0answers
10 views

Understanding Van Dyke's matching rule example

In Hinch's Perturbation Methods book, in the first example of matched asymtoptics example he introduces Van Dyke's matching rule. In the example, he has the equation: $$\epsilon f_{xx}(x) + f_{x}(x) ...
1
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2answers
27 views

What happens to the following limit when $b\in [0, 1)$ and $b > 1$?

Problem: Find all $a, b$ which make the following statement true: $\lim_{x\to 0}\frac{\exp{\left(\sin ax\right)} - \cos x}{x^b} = \frac{1}{2}$ Attempted solution: Firstly, let's notice that if $b ...
0
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3answers
52 views

Find limit: $\lim_{(x,y)\rightarrow(0,0)}{\frac{x^2+2xy+y^2}{x^2+y^2}}$

Can someone help me find this limit: $$\lim_{(x,y)\rightarrow(0,0)}{\frac{x^2+2xy+y^2}{x^2+y^2}}$$. I thought about substituting polar coordinates. But I think thats an overshot. Can someone please ...
3
votes
1answer
59 views

How do I show $\lim_{x\to\infty}f(x) = \lim_{x\to\infty} f '(x)=0$ if $\lim_{x\to\infty}f '(x)^2 + f(x)^3 = 0$? [duplicate]

$f(x)$ is a real valued function on the reals, and has a continuous derivative such that $$\lim_{x\to\infty} f'(x)^2 + f(x)^3 = 0.$$ How do i show that $$\lim_{x\to\infty} f(x) = \lim_{x\to\infty} ...
1
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1answer
19 views

Integrable convex function vanishes at infinity

Why does a function that is Riemann-integrable in $[0, \infty)$ and that is convex vanishes at infinity?
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6answers
165 views

Sum: $1-2+3-4+5-6+…$

If we forget all the rules about infinte sums what am I doing wrong? $$1-2+3-4+5-6+...=\sum_{n=1}^{\infty} n(-1)^{n+1}$$ (with Grandi's series) $$1,1+(-2)=-1,1+(-2)+3=2,1+(-2)+3+(-4)=-2,...$$ we ...
4
votes
4answers
891 views

What kind of growing function has a constant as limit?

My knowledge in mathematics are a bit old and I'm looking for functions with constant as limit. The function must always grow. The curve should be something similar to $\sqrt{x}$ or $\ln(x)$ but with ...
2
votes
3answers
78 views

Show that $\frac{n}{n^2-3}$ converges

Hi I need help with this epsilon delta proof. The subtraction in the denominator as well as being left with $n$ in multiple places is causing problems.
2
votes
4answers
70 views

According to Stewart Calculus Early Transcendentals 5th Edition on page 140, in example 5, how does he simplify this problem?

In Stewart's Calculus: Early Transcendentals 5th Edition on page 140, in example 5, how does $$\lim\limits_{x \to \infty} \frac{\dfrac{1}{x}}{\dfrac{\sqrt{x^2 + 1} + x}{x}}$$ simplify to ...
1
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4answers
149 views

How to find this function's limit?

Let $$ \lim_{x\rightarrow 0} \frac{(x + 1)^{\frac{1}{x}} - e}{x} = ? $$ How would you calculate it's limit? I thought using l'hopital rule, but it then becomes something nasty, as the differentiate ...
0
votes
1answer
69 views

Using the ratio test when the limit of ratio is infinity

If the limit in ratio test is infinity. Does the sequence converge? I suspect not as it is infinity and not some finite value but I'm not sure. Any help?
0
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6answers
122 views

how to prove that $\lim\limits_{n\to\infty} \left( 1-\frac{1}{n} \right)^n = \frac{1}{e}$ [duplicate]

I need to prove $$\lim\limits_{n\to\infty} \left( 1-\frac{1}{n} \right)^n = \frac{1}{e}$$ For now, I only know the e definition: $\lim\limits_{n\to\infty} \left ( 1+\frac{1}{n} \right)^n = e $, and I ...
0
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1answer
40 views

how to solve this formula limits's formula [duplicate]

I have already known one way to solve this formula but I just want to know the easier way to do so: $$\lim_{ u \to 0} \frac{\sin(u)}{u}=1$$ Please kindly help me!! Thank You.
2
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3answers
71 views

Very difficult sequence

How can I show this? I tried with the definiton of the exponential function but it didn't work. $$\displaystyle \lim\limits_{n\to\infty} \frac{(4n^7+3^n)^n}{3^{n^2}+(-9)^n\log(n)}=1$$ I hope ...
2
votes
3answers
44 views

Solving the limit of an integral

Compute $$\lim _{n\to \infty }\left(\frac{\left(4n-4-2a_n\right)}{\pi }\right)^n$$ Where $$a_n=\int _1^n\:\frac{2x^2}{x^2+1}dx$$ The integral I solved and I got $a_n=2(x-\arctan(x))$ Afterwards, ...
2
votes
1answer
56 views

Why are the two limits equal?

I want to show that if $g$ is continuous at $a$ and $f$ at $g(a)$, then $$\lim_{x \to a}{\frac{f(g(x))-f(g(a))}{g(x)-g(a)}} = \lim_{x \to g(a)}{\frac{f(x)-f(g(a))}{x-g(a)}}$$ Now I know that ...
0
votes
2answers
46 views

Verify a property of limits

Is it generally true that for two functions $f$ and $g$, $$\lim_{x \to g(a)}{f(x)} = \lim_{x \to a}{f(g(x))}$$ as long as both exist? This seems to be true when $f$ is continuous at $g(a)$ and $f ...
1
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1answer
35 views

Limit of a function - $x$ either rational or irrational - limit $1$ or $0$. [duplicate]

Show that: The continuous functions $f_{n,k}(x):=(\cos(k!\pi x))^{2n},0\leq x \leq 1$ satisfy the relation $\lim_{k\to \infty}(\lim_{n\to \infty}f_{n,k}(x))=\begin{cases} 1, & \textit{if ...
-1
votes
2answers
38 views

How can the ball reach the wall when it always has to travel halfway? [duplicate]

If I throw a ball at the wall, when it has travelled halfway, it still has half the distance to travel. As it continues, the fraction left to travel continues i.e. one quarter to go, one eighth to go, ...
6
votes
1answer
41 views

Find: $\lim_{x\to0}\left(\lim_{n\to\infty}2^{2n}\left(1-\left(f ^{\circ n}(x)\right)\right)\right)$

Let $$f:[0,1]\to\Bbb R\;\;\mbox{defined by}\;\;\; f(x)=\sqrt{\frac{1+x}{2}}$$ Find: $$\lim_{x\to0}\left(\lim_{n\to\infty}2^{2n}\left(1-\left(\overbrace {f \circ f \circ f \cdot\cdot\cdot \circ ...
5
votes
5answers
131 views

A limit problem: $\lim\limits_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$

I need help in solving the limit below: $$\lim_{n\to\infty}\frac{1+\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} }{1+\frac{1}{3}+\frac{1}{9}+\cdots+\frac{1}{3^n} }$$ What I've done is to simplify ...
3
votes
4answers
103 views

Taking limits on each term in inequality invalid?

So this inequality came up in a proof I was going through. $$c - 1/n < f(x_n) \leq c$$ Where $c$ is a real number, $f(x_n)$ is the image sequence of some arbitrary sequence being passed through a ...
0
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1answer
69 views

Why does the following limit give two answers?

I want to calculate $$ \lim_{t \to 0} \frac{t^2}{\sin^2(t)}$$ and I proceed as follows $$\stackrel{H}{=} \lim_{t \to 0} \frac{2t}{2\sin(t)\cos(t)} \implies \lim_{t \to 0} \frac{2t}{\sin(2t)}$$ ...
5
votes
1answer
95 views

Limit of a sum.

While fixing my answer to this question I noticed that (actually the question is equivalent to this modulo some algebra) $$\frac{1}{2}=\lim_{x\to\infty}\sum_{i=0}^\infty ...
4
votes
3answers
80 views

How do I evaluate $\lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\sum_{k=1}^{n}\frac{1}{\sqrt{k}}$?

How do i evaluate this limit : $$\displaystyle \lim_{n\rightarrow \infty}\frac{1}{\sqrt{n}}\left(1+\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}+............+\frac{1}{\sqrt{n}}\right)$$ ? Thank you for any ...
3
votes
1answer
408 views

Is the Sinc function continuous?

Is $\frac{\sin x}{x}$ a continuous function or is it not? I am confused with the fact that at zero it cannot be defined yet the limit surely exists. So, the question of its continuity arises.
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0answers
58 views

How would you show that $\lim_{n \to \infty} (1+ \frac{1}{n})^n$ is equal to $e$? [duplicate]

How would you show that $\lim_{n \to \infty} (1+ \frac{1}{n})^n$ is equal to $e$?
2
votes
2answers
66 views

why $\lim_{x \to \frac{\pi}{4}} \frac{\cos 2x}{\sin x-\cos x}=-\sqrt{2}$?

I have this very simple limit to find $$\lim_{x \to \frac{\pi}{4}} \frac{\cos (2x)}{\sin x-\cos x}$$ which is equal to $-\sqrt{2}$. However I can get the outcome as mentioned, or $\sqrt{2}$ in the ...
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votes
2answers
46 views

Paradoxical question about infinite bump function. [on hold]

Suppose that you have a bump function of the form $f\left( u \right)=e^{-\left( \frac{1}{1-\left( u \right)^{2}} \right)}$ that is continuous on $\{u$:$\\\ \mid u \mid < 1\}$ Now compose that ...
1
vote
1answer
30 views

Help with a proof about limit point

$\{x_k\}$ is a sequence of real numbers and let $\Omega$ be the set of all distinct points in $\{x_k\}$. Prove that if $\Omega$ has a limit point $x$, then $\{x_k\}$ has a sub-sequence converging to ...
1
vote
2answers
40 views

Evaluting $\lim_{(x,y) \to(0,0)} x \cdot \ln{(x^2+2y^2)}$

How can I find such limit: $$\lim_{(x,y)\to(0,0)} x \cdot \ln{(x^2+2y^2)}$$?
1
vote
2answers
105 views

Limits in multivariable functions

I tried to evaluate this limit but I can't see any limited function here (the limit exists). $\lim\limits_{(x,y)\to(0,0)}\frac{2x^2y}{x^4 + y^2}$ Thank you.
2
votes
1answer
42 views

What is the meaning of $\lim_{\Delta(P) \to 0} F(P) = L$ for partitions

Let $[a,b]$ be an interval, and denote by $\mathcal P[a,b]$ the family of all partitions of $[a,b]$, i.e. sets $P = \{ a = x_0 < x_1 < \ldots < x_n = b \}$. For some $P \in \mathcal P[a,b]$ ...
1
vote
4answers
84 views

$ \lim_{x \rightarrow \infty} e^{1/x} = a $ is not equivalent with $ \lim_{x \rightarrow \infty} a^x = e$?

I have problems understanding, why $$ \lim_{x \rightarrow \infty} e^{1/x} = a $$ is not equivalent with $$ \lim_{x \rightarrow \infty} a^x = e. $$ In the first case there is a solution $a=1$, and ...
4
votes
1answer
49 views

Limit of equations.

$a_{n}+b_{n}+c_{n}=2n-1 $ $a_{n}b_{n}+b_{n}c_{n}+a_{n}c_{n}=2n+1 $ $a_{n}b_{n}c_{n}=-1 $ $a_{n}<b_{n}<c_{n}.$ Find $\lim\limits_{n\to \infty}na_{n}.$ ATTEMPT: From $ (2). $ ...
3
votes
2answers
206 views

How to prove the non-existence of a polynomial series that uniformly converges to this function?

I was asked to prove the following. There exists a function $p(x)\in C(a,b)$ $(-\infty<a<b<+\infty)$, such that there does not exists a polynomial series which uniformly converges to ...
10
votes
1answer
90 views

Closed form for $\sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+…}}}$

Is there a close form for the following nested radical? $$\sqrt{1+\sqrt[2!]{2^2+\sqrt[3!]{3^3+...}}}$$ It converges and $$\quad \quad \lim_{n \to\infty} ...
2
votes
1answer
51 views

About the computation of the limit $ \lim_{x \to \infty} \frac{1^{99} + 2^{99} + \cdots + x^{99}}{x^{100}} $

I was reading this post and I don't understand why I can't do this: \begin{align*} \lim_{x \to \infty} \frac{1^{99} + 2^{99} + \cdots + x^{99}}{x^{100}} &= \lim_{x \to \infty} ...