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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.

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2answers
18 views

How to find the limit of indeterminate form

Find the limit of the given expression $\lim_{x\to\frac{\pi}{2}}\left ( \sec x-\frac{1}{1-\sin x} \right )$
-1
votes
3answers
35 views

Why is $\sqrt{(\cos^2 \phi + \sin^2\phi)} = 1$?

A rather short question: Why is $$\sqrt{(\cos^2 \phi + \sin^2\phi)} = 1$$ I have seen that in
0
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1answer
26 views

Does L'Hopital's Rule extend to $x \rightarrow \infty$ and $L= \infty?$

The following is given: Suppose that $f$ and $g$ are continuous functions on the interval $(a,b)$ and that $$\lim_{x \rightarrow b} \frac{f(x)}{g(x)} = \frac{0}{0}.$$ If $g(x),g'(x) \neq 0$ for all $...
0
votes
2answers
50 views

Tricky limit algebra

Could someone please help me figure out the following algbera? I really don't understand any of the steps. $$ \begin{align*} \lim_{x\to 0} \frac{1-\sqrt{1-x^2}}{x} &= \lim_{x\to 0} \frac{(1-\...
-1
votes
0answers
12 views

Prove $L$ is in $\mathbb {Z} $ - Limits by definition

Let $a, b \in R$ with $a<b$ and $f : (a, b) \to \mathbb {Z}, x_0 \in (a, b) $. Suppose that there exists an $L\in\mathbb{R}$ such that $\lim_{x\to x_0} f(x) = L $. Prove that $L \in \mathbb{Z} $.
1
vote
1answer
17 views

Sequence with limit $\infty$

Let $x_n$ be a sequence of real strictly positive numbers. If $\lim_{n\to \infty} x_n=\infty$, is it necessary for the sequence to be strictly increasing? If not, give a counterexample. Intuitively I ...
1
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3answers
40 views

Evaluating trigonometric limits with variable exponent $\left(\frac{\tan x}{x}\right)^\frac1x$

I found a set of problems of limits which i can't seem to work my way around. I tried using the natural log and then applying L'Hospital's rule but I can't seem to make it work. The problem was to ...
0
votes
1answer
48 views

Calculate $\lim_{n\to\infty}x_n$ when $x_{n+1} = \frac{1}{4}x_n(5-x_n)$

Sequence $x_n$ such that $x_0 \in [0, 2]$ is defined by $x_{n+1} = \frac{1}{4}x_n(5-x_n)$. For what values of $x_0 \in [0, 2]$ does $x_n$ converges and to what limit? For $x_0 = 0$, $\lim_{n\to\infty}...
0
votes
2answers
31 views

Prove function is differentiable at a point

Prove that the function $$f(x) = \left\{ \begin{array}{ll} x^2, & x \in \mathbb{Q} \\ 0, & x \in \mathbb{Q}^c \\ \end{array} \right.$$ is differentiable at $x = 0.$ I'm not ...
0
votes
2answers
39 views

A difficulty in understanding a proof for L'Hospital's rule (in Petrovic)

The theorem and its proof is given below: But I could not understand why $F$ & $G$ are defined as thought, could anyone explain this for me please?
1
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3answers
65 views

Finding the limit of $\frac {e^{-1/x^2}}{x^{100}}$ as $x \rightarrow 0$

Finding the limit of: $$\frac {e^{-1/x^2}}{x^{100}}$$ as $x \rightarrow 0$ My answer is: 1- I can not apply L`hopital rule because the limit will still be 0/0 because the numerator will still be ...
2
votes
2answers
46 views

Finding the limit $\frac {\sin x - \arctan x}{x^2 \ln x}$ as $x \rightarrow 0$ in Petrovic

Finding the limit: $$\lim_{x \to 0}\frac {\sin x - \arctan x}{x^2 \ln x}$$ My questions: 1- I think the question should be corrected to as $x \rightarrow 0^+$, because of the domain of $\ln x$ .......
0
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4answers
56 views

Using the epsilon-delta definition, prove that the $\lim_{x \rightarrow 2} \frac{2x}{x-3}=-4$

Here is what I have so far, and I am new to these kinds of proofs so please be detailed with me: $| \frac{2x}{x-3}-(-4) | = |\frac{6(x-2)}{x-3}|$ But I don't understand what I am supposed to do with ...
0
votes
2answers
27 views

Find all $\alpha$ such that $a_n = n^\alpha(\sqrt [n^2] a - 1)$ converges

Find all $\alpha$ such that the sequence $a_n = n^\alpha(\sqrt [n^2] a - 1)$ converges to a non-zero number and calculate this number. Assume $a \neq 1$ is positive. I have no idea where to start ...
5
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2answers
75 views

How to find $\lim_{n\to\infty}(pa_n + qb_n)^n$ with $p + q = 1$?

Suppose $a_n$ and $b_n$ are sequences of positive numbers, such that $$ \lim_{n\to\infty}a_n^n = a,\quad \lim_{n\to\infty}b_n^n = b,\qquad a,b\in (0, \infty). $$ Find limit $$ \lim_{n\to\infty}...
1
vote
0answers
46 views

Why is (or was) it necessary to rigorously define the concept of a limit?

I've been revisiting calculus for the fun of it and trying to "reconstruct" over history to understand what forces led to the creation of various topics. The reason for rigorously defining limits is ...
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votes
2answers
34 views

find the limit $\lim _{h\to 0} [ \sin^4( \pi/4 +h) − \sin^4( \pi/4 )] / h$. [on hold]

The mark scheme said it was 1 but I'm sure it has to be 0 but now it's getting confusing.
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votes
1answer
32 views

Limit of $\cos(1/x^2)$ as $x$ tends to $0$

$\displaystyle\lim_{x\to0}\cos\frac{1}{x^2}$ I understand that through the squeeze theorem we know the upper bound is $1$ and lower bound is $-1$ I understand that it would be undefined at $0$ and ...
1
vote
1answer
65 views

Limit of $\frac{\tan{(\sin{(x)}})}{\sin{(\tan{(x)}}}$ when x approaches 0

How would one approach finding this limit without using Taylor's series? $$\lim_{x \to 0} \frac{\tan{(\sin{(x)}})}{\sin{(\tan{(x)}})}$$
0
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1answer
22 views

Suppose $f(x)=g(\frac{1}{x})$ for $x\gt{0}$, and let $L\in{\mathbb{R}}$. Prove that $\lim_{x\to \infty}f(x)=L$ iff $\lim_{x\to 0^+}g(x)=L$.

I know this is a iff statement so I will assume each side and then try to prove the other. But I do not know how to actually prove this. I know if I assume $\lim_{x\to \infty}f(x)=L$ then $\lim_{x\to \...
0
votes
1answer
25 views

Limit of the form infinity plus infinity: $\lim\limits_{x\to1} \left(\frac1{x-1}+\frac2{1-x^2}\right)$ [on hold]

Lim(1/(x-1) + 2/1-x^2) X->1 $$\lim_{x\to1} \left(\frac1{x-1}+\frac2{1-x^2}\right)$$ Is this limit in an undetermined form? I guess that its result is +...
1
vote
2answers
64 views

How to calculate $ \lim_{x\to 4} (\frac{\frac{\pi}{6} - \arcsin(\frac{\sqrt{x}}{4})}{\sqrt[3]{2x-7}-1}) $ without the rule of L'Hôpital?

$$ \lim_{x\to 4} (\frac{\frac{\pi}{6} - \arcsin(\frac{\sqrt{x}}{4})}{\sqrt[3]{2x-7}-1}) $$ Hello! I need to solve this limit. I had solved it with the rule of L'Hôpital, but i can't without it. I ...
0
votes
3answers
25 views

can we separate out absolute value function like this?

$$ \lim_{x\to 2^-}\frac{x(x-2)}{|(x+1)(x-2)|}= \lim_{x\to 2^-}\left(\frac{x}{|x+1|}\cdot \frac{x-2}{|x-2|}\right) $$ So as the title says, is it okay to separate function under absolute value like ...
1
vote
3answers
37 views

How to prove that the sequence $a_n = (1/n)^{1/n}$ converges?

Using a calculator I can see that the sequence $$ a_n =\left(\frac{1}{n}\right)^{\frac{1}{n}} $$ converges to $1$ as $n$ approaches infinity. I would like to know the correct way to prove this. It ...
0
votes
3answers
88 views

Given two convergent sequences find $\lim_{n\to\infty}\sqrt[n]{a_n^n+b_n^n+2018}$

I tried to solve this question, but can't get to a correct answer... Let $a_n,b_n$ be two sequences s.t. $$ a_n\xrightarrow{n\to\infty}a>1,\quad b_n\xrightarrow{n\to\infty}b>1. $$ Find ...
1
vote
0answers
48 views

Prove $a_n$ and $b_n$ have the same limit [duplicate]

Let x,y be positive, constant nambers so: $a_1=x, b_1=y$ and $a_{n+1}=(a_n+b_n)/2,\; b_{n+1}=\sqrt{a_nb_n}$ Prove that $a_n$ and $b_n$ have the same limit. From inequality of arithmetic and ...
1
vote
1answer
57 views

Where is $\sqrt{z+1}$ analytic and continuous?

I am trying to determine where $$f(z)=\sqrt{z+1}$$ is analytic, where the square root is the principal branch. I know that $\sqrt{w}$ is analytic for $\mathbb{C}\setminus(-\infty,0]$. So, I think $f(...
0
votes
1answer
20 views

Prove that every unbounded sequence contains a subsequence that tends to inifnity

I know the issue has already been addressed in other posts, but I want to ask if the proof I constructed is correct - I'm not confident with proofs. If the sequence (xn)n is unbounded, then by ...
0
votes
1answer
39 views

Limit of integral sequence

I am asked to prove that $$\lim_{n\to\infty}\int_0^{\frac{\pi}{2}}\sin(n^4\cos t)dt=0$$ I think you have to calculate $$\int_0^{\frac{1}{n}}\sin(n^4\cos t)dt$$ and $$\int_{\frac{1}{n}}^{\frac{\pi}{2}}\...
2
votes
4answers
52 views

Is the limit of a strictly increasing function always $\infty$?

If $f(x)$ is strictly increasing, is $\lim_{x\to\infty} f(x) = \infty$ ? Also is $\lim_{x\to -\infty} = 0?$ I think the answer is yes. A good example is $e^{x}$. I don't know how to show this claim ...
5
votes
2answers
41 views

Other ways to find a limit where the denominator produces $0$, besides factoring and cancellation?

So I came across a seemingly innocent looking integral: $$\int_{-2}^1 \frac{1}{x^2}dx$$ Now of course, when taking the antiderivative then plugging in the values, we can see that we get a nonzero/$0$ ...
0
votes
4answers
41 views

Limit for $x\to\pi$ for the trigonometric function $(1+\cos x)/\tan^2x$

$\displaystyle \lim_{x\to\pi}\frac{1+\cos x}{\tan^2x}$ How would I be able to solve this without using L'Hôpital's rule? I have tried everything with normal identity manipulation etc and I have ...
0
votes
3answers
77 views

When does $\sum_{n=1} ^{\infty} (\sqrt [n]x -1)$ converge?

$A=\{ x| \sum_{n=1} ^{\infty} (\sqrt [n]x -1) \ \text{is convergent \}}$ . Now what is $A$ ? 1- $\{1\} $ 2- $(0, \infty)$ 3-$(0,1]$ 4-$(\frac{1}{e} , e) $ It is clear that $1 \in A$ and $\lim _{n ...
0
votes
2answers
41 views

What happen's to $\epsilon_1(x), \epsilon_2(x)?$

Let $f: (a,b) \rightarrow \mathbb{R}$ be given. If $f''$ exists, prove that $$\lim_{h \rightarrow 0} \frac{f(x-h)-2f(x)+f(x+h)}{h^2} = f''(x).$$ Find an example that this limit can exist even when $f''...
2
votes
3answers
36 views

Hints about the limit $\lim_{x \to \infty} ((1+x^2)/(x+x^2))^{2x}$ without l'Hôpital's rule?

I've tried to evaluate $\lim_{x \to \infty} \left(\frac {1+x^2}{x+x^2}\right)^{2x}$ as $$\lim_{x \to \infty} \left(\left(\frac {1+ \frac{1}{x^2}}{1+ \frac{1}{x}}\right)^{x}\right)^{2}$$ So the ...
0
votes
3answers
50 views

how to calculate $\lim_{x\to 0} \frac{3^x-1}{x}$ [on hold]

What is the limit $\lim_{x\to 0} \cfrac{3^x-1}{x}$ without using l'hopital. Can you give me some clue or advice? I believe there is euler limit invlved.
1
vote
1answer
50 views

How to calculate $ \lim_{n\to\infty}\prod_{i=0}^n (1 - \frac{1}{3i+2})$?

How to calculate $$ \lim_{n\to\infty}\prod_{i=0}^n (1 - \frac{1}{3i+2})\quad? $$ By transforming it with natural logarithm, the product is equal to $$ \large e^{\sum_{i=0}^n\ln(1-\frac{1}{3i+2})}...
0
votes
4answers
72 views

Why is $\lim_{n\to \infty} \frac{n^{4n}}{(4n)!} = 0$?

I'm trying to figure out how to prove, that $$\lim_{n\to \infty} \frac{n^{4n}}{(4n)!} = 0$$ The problem is, that $$\lim_{n\to \infty} \frac{n^{n}}{n!} = \infty$$ and I have no idea how to prove the ...
4
votes
1answer
47 views

Prove that $\prod_{k=2}^{+\infty} (1+1/k^2) = \sinh(\pi)/(2 \pi)$.

My attempt 1: Let $x_n=\left(1+\frac{1}{2^2} + \cdots + \frac{1}{n^2} \right)$, and we have $x_{n+1}>x_n$. Since $$ 1+\frac{1}{n^2} \le 1+ \frac{1}{(n-1)(n+1)} = \frac{n}{n-1} \cdot \frac{n}{n+1}, $...
1
vote
2answers
69 views

What is $\lim_{(x,y)\to(0,0)}\frac{ (x^2y^2}{(x^3-y^3)}$?

I've been trying to solve $$\lim_{(x,y)\to(0,0)} \frac{(x^2y^2)}{(x^3-y^3)}$$ for a while and can't get anywhere with it, is it possible to solve this using the squeeze theorem?
1
vote
1answer
28 views

$\max\{a_1,a_2,\dots,a_n\}$ converges for a convergent sequence $a_n$

I am tackling the following question and want to be sure that my reasoning is fine. Let $a_n$ be a convergent sequence s.t $\displaystyle \lim_{n\to\infty}a_n=a$. Let $$b_n\triangleq\max\{a_1,a_2,\...
0
votes
2answers
32 views

Limits of product of function and measure

Let $(X, \mathbb A, m)$ be a measurable space and $f: X \to \mathbb R$ a Borel-measurable function. If $\psi: [0, + \infty) \to [0, + \infty)$ is monotone non decreasing and $\int \psi ( |f|) dm < +...
3
votes
2answers
37 views

How to evaluate $\lim_{n\to\infty} \dfrac{1^p+2^p+…+n^p}{n^p}-\frac{n}{p+1}$? [duplicate]

$$\lim_{n\to\infty} \dfrac{1^p+2^p+...+n^p}{n^p}-\frac{n}{p+1}\text{ with } p\in\mathbb N$$ I don't really know how to start, I mean... I could try substituting by $\Big(\dfrac{n(n+1)}{2}\Big)^p$ ...
-2
votes
4answers
47 views

$\lim_{x\to -\infty}\sqrt{(x^2+2x)}-\sqrt{(x^2-2x)}$ [on hold]

I have no idea how to attack and solve this problem. The solutions manual shows the answer is $-2$, but how do I get there? $$\lim_{x\to -\infty}\sqrt{(x^2+2x)}-\sqrt{(x^2-2x)}$$
-2
votes
2answers
39 views

How can I find the limit $\lim_{x\to \frac12}\frac{4x^2 - 1}{\arcsin(1 - 2x)} $ [on hold]

How can I find the limit $$\lim_{\left(x\to 1/2\right)}\ \frac{4x^2 - 1}{\arcsin(1 - 2x)}\quad ? $$
0
votes
2answers
34 views

Limit as $z$ approaches $0$ for $e^{1/z^4}$ [duplicate]

I want to work out if the limit as $z$ approaches $0$ for $e^{1/z^4}$ exists and if not then why. I worked out that the left and right sided limits both equal to +∞ so I thought that was enough to ...
2
votes
2answers
20 views

Determine pointwise and uniform convergence of a given sequence of functions and calculate an integral

Let $f_n:[0,\infty)\to\mathbb{R},f_n(x)=\frac {ne^{-x}+xe^{-n}}{n+x},\space\space\space\forall n\in\mathbb{N}$. Study the convergence and calculate: $A_n=\int_0^1f_n(x)dx.$ My attempt: For the ...
2
votes
0answers
35 views

Limit of a recursive sequences involving the AM, GM, and HM (arithmetic-geometric-harmonic mean)

Let $x,y,z$ be positive real numbers. And let $\text{AM}$, $\text{GM}$, $\text{HM}$ respectively be the arithmetic mean, geometric mean, and harmonic mean. Define $$a_n=\text{AM}(a_{n-1},g_{n-1},h_{...
0
votes
1answer
17 views

Prove that the limiting probability of a transient state in a Discrete Time Markov Chain, is 0

I've got this theorem that we covered in class, but I didn't get a chance to write down the proof. I'm not quite sure how to prove it. I'm not sure how to go about proving this on my own, and I can't ...
0
votes
1answer
26 views

Find the limit of the function $f(x) = \frac{1}{1+e^{-m(x-x_0)}}$ as $x\to+\infty$ and $x\to-\infty$.

Give the limits of $f(x) = \frac{1}{1+e^{-m(x-x_0)}}$ as $x\to+\infty$ and $x\to-\infty$. It's a Sigmoid function, so I know to expect integer limits. To start I set the $\lim_{x\to\infty} f(x) = L,...