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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.

3
votes
3answers
57 views

How to calculate this limit without L'Hopital rule?

I want to evaluate the following limit without using the L'Hopital rule : $$ \lim\limits_{x\rightarrow 0^+}\frac{e^{x\ln(x)}-1}{x}$$ I know the answer is $-\infty$. I can demonstrate that graphically ...
0
votes
1answer
13 views

Factor square root out of quotient

How do I get from the first expression to the second? The only reason the limits are included is because on WolframAlpha it mentioned that this was the case for large negative numbers of x: $\lim_{x\...
-1
votes
3answers
53 views

Calculating $\lim_{x \to 0} \frac{\ln \sin(x)}{\ln\tan(x)}$

I need some help calculating this limit: $$\lim_{x \to 0} \frac{\ln \sin(x)}{\ln \tan(x)}$$
0
votes
2answers
19 views

Proof: All directional derivatives $\frac{\partial f}{\partial e}$ of $\frac{sin(x^3+y^3)}{x^2+y^2}$ are in the origin

Let $M := (0,\infty) \subset \mathbb{R^2}$ and $f:\mathbb{R}^2 \to \mathbb{R}$. How can one prove that all directional derivatives $\frac{\partial f}{\partial e}$ of $f(x,y)$ are existing in the ...
1
vote
1answer
41 views

Using $\epsilon-\delta$ language to show $\lim_{x\rightarrow0}\frac{1-\cos(x)}{x}=0$.

I've permitted myself to use $\displaystyle\lim_{x\rightarrow0}\frac{\sin(x)}{x}=1$, as I've proven this. Here's some manipulation I've tried, but I can't quite get the $1+\cos(x)$ terms in the ...
0
votes
1answer
46 views

Evaluating $\lim_{n\to\infty}\sqrt[n]\frac{\prod_{k=1}^{n}(2n+k)}{n^n}$

How to find $$\lim_{n\to\infty} \sqrt[n]\frac{\prod_{k=1}^{n}(2n+k)}{n^n}\quad ?$$ Wolfram Alpha says that it's $\frac{27}{4{e}}$, but I fail to arrive at this result. Methods allowed in my class ...
0
votes
1answer
21 views

Show that the sequence $a_1=3$ and $a_{n+1}=\frac{3(1+a_n)}{3+a_n}$ converges to $\sqrt3$

First of all let´s see how it goes: $3,\frac{3(1+3)}{3+3}=\frac{12}{6}=2,\frac{3(1+2)}{3+2}=\frac{9}{5}=1.8,...$ We see that it's decreasing. What i need to show first is that the sequence decreases....
0
votes
3answers
47 views

Find a function $f:\mathbb R\to\mathbb R$ s.t $f(a)+f(b)=f(a)$ $\forall a>b$, $f(a)>f(b)$ $\forall a>b$.

Define a function $f$ that satisfies: $f(a)+f(b)=f(a)$ $\forall a>b$ and $f(a)>f(b)$ $\forall a>b.$ The answer says such function does not exist but does not say why. Let $f(x)=\lim_{\...
-1
votes
2answers
58 views

How to prove that $f(x)=\lim_{a\to\infty}e^{-ax}$ is not a function?

$y=\lim_{a\to\infty}e^{-ax}$. How to prove that there does not exist a function $f$ such that $f(x)=y$?
0
votes
1answer
14 views

$f(x) = 1 - |1 - 2x|$, $a_{n+1} = f(a_n)$, prove convergence of subsequences

Let $f(x) = 1 - |1 - 2x|$, $a_1 = a$, $a_{n+1} = f(a_n)$. Prove there exists $a \in [0, 1]$ such that for every $x \in [0, 1]$ there exists a subsequence of $a_n$ convergent to $x$. I've tried to ...
0
votes
4answers
59 views

Help to show $\lim_{k\to\infty} \frac{2^kk!}{\sqrt{(2k+1)!}}=0$

Can you help me to show that: $$\lim_{k \to \infty} \frac{2^kk!}{\sqrt{(2k+1)!}}=0$$
3
votes
4answers
74 views

Evaluate $\lim \limits_{n \to \infty\ } \Biggl( \frac{2,7}{(1+\frac{1}{n})^n}\Biggr)^n=0?$

$\lim \limits_{n \to \infty\ } \Biggl( \frac{2,7}{(1+\frac{1}{n})^n}\Biggr)^n$ I would like to replace $(1+\frac{1}{n})^n$ by $e$, and then $\frac{2,7}{e}<1$, so $\lim \limits_{n \to \infty\ } \...
4
votes
1answer
74 views

$\lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!} = \frac{1}{2}$ - basic methods

Prove that $\lim\limits_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!} = \frac{1}{2}$. This question was asked a several times on MSE, but each time it was solved using Poisson distribution or a ...
2
votes
2answers
36 views

Limit of a sequence with n root

I have a trouble with this example: $$n(\sqrt[n]n-1)$$ I've been trying to do it this this way: $$a_n = \frac{(\sqrt[n]n-1)(\sqrt[n]n^{n-1}+\sqrt[n]n^{n-2}+\dots+1)}{\sqrt[n]n^{n-1}+\sqrt[n]n^{n-2}+...
3
votes
7answers
62 views

Help calculating $\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$

I need some help calculating this limit: $$\lim_{x \to \infty} \left( \sqrt{x + \sqrt{x}} - \sqrt{x - \sqrt{x}} \right)$$ I know it's equal to 1 but I have no idea how to get there. Can anyone give ...
0
votes
2answers
65 views

$(I-A)^{-1}=\sum_{i=0}^\infty A^i$

Let $V$ be a finite dimentional normed vector space and let $A$ a linear transformation from $V$ to $V$ such that $\left \| A \right \|<1$. Show that the linear tranformation $I-A$ is invertible ...
0
votes
3answers
31 views

Limit of $\lim_{x \to +\infty} x \left( \sqrt{(1+\frac{a}{x}) (1+\frac{h}{x})} -1 \right)$

Is there a way to calculate this limit? $$\lim_{x \to +\infty} x \left( \sqrt{(1+\frac{a}{x}) (1+\frac{h}{x})} -1 \right)$$ I know it's equal to $\frac{a+h}{2}$, but what method can I use to ...
1
vote
1answer
43 views

Evaluate $\lim \limits_{n \to \infty\ } \sqrt[n]{\left|\frac {1}{n3^n}-\frac {n^{170}}{4^n}\right|}$

$\lim \limits_{n \to \infty\ } \sqrt[n]{\left|\frac {1}{n3^n}-\frac {n^{170}}{4^n} \right|}= \ldots=\lim \limits_{n \to \infty\ } \sqrt[n]{\frac {1}{n3^n}} \cdot\sqrt[n]{\left|1-\left(\frac {3}{4}\...
1
vote
3answers
41 views

Is there a more general definition of a limit?

I've been bothered with the standard definition of the limit in analysis for a while: 'A sequence $(x_n)$ has $x$ as a limit if $\forall \epsilon>0\ \exists N\in\mathbb{N}$ s.t. $\forall n\geq N:\ ...
1
vote
4answers
37 views

Prove that a sequence is divergent (By definition - Epsilon-N Way)

First, this is the question: Prove (using epsilon-N definition) that the sequence $ a_n = \left<\sqrt{n}\right> $ is divergent. Note: $ \left<x\right> = x- \lfloor x \rfloor$ My ...
1
vote
1answer
28 views

Find the limit of complex number exponentiation

How do I find the limit of $$ Z_n=\left(1+{\frac{a+bi}{n}}\right)^n $$ Should I have real and imaginary parts? Like this $$ \lim Z_n = \lim \left(\left(1+{\frac{a}{n}}\right)+{\frac{b}{n}}\right)^n ...
0
votes
2answers
63 views

How to calculate $\lim_{x\to 0} \frac{\sin{(\pi \sqrt {x+1} )}}{x}$ without L'Hospital's rule?

I got stuck trying to calculate the following limits: $$ \lim_{x\to 0-} \frac{\sin{(\pi \sqrt {x+1} )}}{x},\quad \lim_{x\to 0+} \frac{\sin{(\pi \sqrt {x+1} )}}{x} $$ I've tried to approximate the ...
-5
votes
5answers
33 views

calculate limit $x\rightarrow0^-$ without l'hospital [on hold]

$\displaystyle\lim_{x\rightarrow 0^-} \left(\frac{\pi \sqrt {x+1} }{x}\right)$ I can't move with it, I tried using 3-functions but it didn't help. edit: the (x+1) is under square root
1
vote
2answers
79 views

Show that $\lim\limits_{x \to \infty}\frac{\ln x}{x}=0$ from definition.

Show that $\displaystyle\lim_{x \to \infty}\frac{\ln x}{x}=0$. I know this is a simple application of the L'Hopital's rule, but can we also show this from the $\displaystyle\epsilon-\delta$ ...
0
votes
2answers
27 views

Prove that a sequence has no limit

Given $Z_n=\arg{\frac{i^n}{n}}$, how do I show that it has no limit?
1
vote
4answers
45 views

Limit of sequence: $\lim_{n\to\infty}({2n+1\over 3n})^n$

$\lim_{n\to\infty}({2n+1\over 3n})^n$ For sufficiently large values of $n$: $0\le ({2n+1\over 3n})^n\le({2\over 3})^n$, From Squeeze theorem: $\lim_{n\to\infty}({2n+1\over 3n})^n=0$ Is it ok? I ...
-1
votes
1answer
25 views

Limit of The Dirac Comb

The Dirac comb function with period T is: $$ f(t,T):=\sum_{k=-\infty}^{k=\infty}\delta(t-kT) $$ What is the limit of: $$ \lim_{T\to0} f(t,T) $$ ?
3
votes
3answers
71 views

How to evaluate $\lim_{n \to \infty} \frac{(1+\sqrt 2)^n+(1-\sqrt 2)^n}{(1+\sqrt 2)^n-(1-\sqrt 2)^n}$?

Evaluate $$\lim_{n \to \infty} \frac{(1+\sqrt 2)^n+(1-\sqrt 2)^n}{(1+\sqrt 2)^n-(1-\sqrt 2)^n}.$$ I tried to expand using Newton's Binomial Theorem, but it didn't work.
0
votes
1answer
50 views

Show that the natural density is $1/2$.

Let $$A_b= \left\lbrace{p \in \mathbb{P}| \left(\frac{b}{p}\right)=1 } \right\rbrace $$ and $$ \nu(A_b)=\lim\limits_{x \to \infty} \frac{\#\lbrace {p \in A_b|p\le x}\rbrace}{\pi(x)}$$ the natural ...
1
vote
2answers
25 views

Finding the minimum of function including a $-\frac{1}{x}$ term.

I would like to find the global minimum of the function: $$f(x, y) = 10(y^2-2x^3)^2 + (1-x)^2 - (y-1000)^{-1}.$$ Now my problem is the following. I know that $(1, \sqrt{2})$ is a local minimum, and I ...
1
vote
3answers
29 views

How to find $\lim_{n\to \infty} (\frac {n}{\sqrt{n^2+n}+\sqrt{n}})$?

I am trying to solve this : $\lim_{n\to \infty} (\frac {n}{\sqrt{n^2+n}+\sqrt{n}})$ but I always end up with $\frac {\infty}{\infty}$ which is undefined I tried for eg $$\lim_{n\to \infty} (\frac {n}{...
0
votes
1answer
50 views

How to analyze $f(f(x))=-x^3+\sin(x^2+\ln(1+\left|x\right|))$?

Define $f\in C^{0}\left(\mathbb{R}\right)$ satisfying $f(f(x))=-x^3+\sin(x^2+\ln(1+\left|x\right| ))$. Prove that this equation has no continuous solution. To figure out the proof, I thought like ...
1
vote
0answers
27 views

How can we show that the limit of the following surface integral is finite?

I have an arbitrary parameterized surface $S(u,v)$ in Cartesian coordinates (with finite area) and the surface passes through origin. There is a very small circular hole of radius $\epsilon$ at the ...
0
votes
2answers
25 views

A differentiable function with itself and its derivative converge to constants, can we conclude its derivative converge to zero [duplicate]

I have been struggled with the following problem. Suppose that $f$ is differentiable on $(0, \infty)$. If we have $\lim_{x \rightarrow \infty} f(x) = c_1$ and $\lim_{x \rightarrow \infty} f'(x) = c_2$...
0
votes
6answers
54 views

Limits - Calculating $\lim\limits_{x\to 1} \frac{x^a -1}{x-1}$ where $a \gt 0$ without using L'Hospital's rule

I'm messing around with this limit. I've tried using substitution for $x^a -1$, but it didn't work out for me. I also know that $(x-1)$ is a factor of $x^a -1$, but I don't know where to go from ...
-3
votes
2answers
20 views

Find the limit of sequence (complex numbers)

How do I find the limit of $$Z_n=n\sin{\frac{i}{n}}$$ I'd say it is 0, but the book says $i$.
2
votes
3answers
261 views

How to compute $(-1)^{n+1}n!(1-e\sum_{k=0}^n\frac{(-1)^k}{k!})$?

I was doing some work on the integral $\int_0^1 x^ne^x dx$ and I eventually came to this expression in terms of n $$\int_0^1 x^ne^x dx=(-1)^{n+1}n!\biggl(1-e\sum_{k=0}^n\frac{(-1)^k}{k!}\biggr)$$ Now ...
2
votes
2answers
144 views

Limit to compare growth of function

I wanted to compare growth of two functions $F_1:n^{\,\lg\,\lg n}$ $F_2:(3/2)^n$ $\lim_{n \to \infty} \frac{n^{\lg\lg n}}{(3/2)^n}$ After differentiating it $\lg \, \lg n$ times I get $\lim_{n \...
1
vote
1answer
21 views

Find the limit of complex number

I have $$Z_n = e^{-i({\frac{\pi}{2}+{\frac{1}{2n}}})}$$ Therefore, as $n \to \infty$, $$\operatorname{\lim}Z_n = e^{-i{\frac{\pi}{2}}}$$ But the answer on the book is $i.$
0
votes
3answers
30 views

Convince me: limit of sum of a constant is infinity

So I have a problem and have simplified the part I am confused about below. If $\sum_{m=1}^{\infty }c < \infty$ and $0 \leq c \leq 1$, then $lim_{n\rightarrow \infty} \sum_{m=n}^{\infty }c= 0$ ...
0
votes
3answers
44 views

Evaluate limit $a^xx^a$ where 0<a<1

I want to evaluate the limit:$$\lim_{x \to \infty}{x^aa^x} $$ where $0<a<1$ I put in Wolfram Alpha and I get the limit is zero We see if the base is $0<a<1$ then : $$\lim_{x \to \infty}{...
0
votes
3answers
29 views

Limit Query on $-\infty$ or $\infty$ answer

Let $f(x)=x+\log_e(x)-x\log_e(x)$ I am confused if I want to know whether $\lim_{x\rightarrow \infty }f(x)$ is $-\infty$ or $\infty$ I used the following concept $f(x)=x+\log_e(x)-x\log_e(x)$ $\lim_{...
2
votes
4answers
40 views

$\lim _{x\to \:y}\left(\frac{x^{n+1}-y^{n+1}}{x-y}\right)=y^n\left(n+1\right)$

try, without L Hopital $$\lim _{x\to \:y}\left(\frac{x^{n+1}-y^{n+1}}{x-y}\right)=y^n\left(n+1\right)$$ try using simple algebra, can not be derived, using only alegebraic tricks.
-4
votes
2answers
44 views

How to compute $\lim\limits_{n \to \infty} \frac{n2^n}{e^n}$? [on hold]

How to solve this question, even when i am applying l hospital rule i am not getting the answer ?
0
votes
1answer
48 views

When is $\int_{-\infty}^\infty f(x) dx = \lim\limits_{n \to \infty} \int_{-n}^n f(x) dx$?

When is $\int_{-\infty}^\infty f(x) dx = \lim\limits_{n \to \infty} \int_{-n}^n f(x) dx$? Normally, we would need to take two different limits, but I am wondering if there is necessary and/or ...
-1
votes
0answers
16 views

Tips for Multivariable Calculus [on hold]

I have a final exam for multivariable calculus coming up and I ask for some tips on these topics: finding the bounds on a triple integration problem (especially phi in spherical) proving a limit ...
-5
votes
0answers
39 views

Show - without using L'Hospital Rule - that $\lim_{x\to 0} \frac{\tan x-\sin x}{\sin x-x}=3$ [on hold]

Show - without using L'Hospital Rule - that $$\lim_{x\to 0} \frac{\tan x-\sin x}{\sin x-x}=3$$ Thanks beforehand
14
votes
2answers
199 views

Does this sequence $a(n) = \frac{1}{n^3\sin(n)}$ converge

Does the sequence $$a(n) = \frac{1}{n^3\sin(n)}$$ converge ? I tried all possible standard calculus approaches but to no avail ... edit: I tried using the root theorem and the limit of the $\frac{...
1
vote
2answers
49 views

$L= \lim_{n \to \infty}\sqrt{n}\int_0^1\dfrac{dx}{(1+x^2)^n}$ [duplicate]

Suppose the limit $$L= \lim_{n \to \infty}\sqrt{n}\int_0^1\dfrac{dx}{(1+x^2)^n}$$ and is larger than $\frac{1}{2}$. Then (A) $0.5<L<2$ (B) $2<L<3$ (C) $3<L<4$ (D) $L\ge 4$ I ...
0
votes
2answers
33 views

$\sum_{k=1}^{n} \arccos k$

Compute $\lim_{n \to \infty} \sum_{k=1}^{n} \arccos k$. My attempt :the sequence is strictly increasing and if I prove that it is not upper bounded its limit is $\infty$. However, I can't prove the ...