Questions on the evaluation and properties of limits.

learn more… | top users | synonyms

1
vote
5answers
30 views

How to prove that $\lim\limits_{n\to\infty}\sqrt[n]{p}=1$?

Could you tell me how to show if $p>0$ then$\lim\limits_{n\to\infty}\sqrt[n]{p}=1$? (+clues) 1.put $\sqrt[n]{p}=1+h_{n}$ 2.Bernoulli's inequality If you don't mind, use the clues to prove it.
1
vote
1answer
28 views

Why is this piece-wise limit equal to 2?

$$f(x) = \begin{cases} 2x-2, & x < 3 \\ 2x-4, & x \ge 3 \end{cases} $$ Why is $$\lim\limits_{h \to 0^+} \frac{f(3+h)-f(3)}{h} = 2 ??$$ Note the (+) in the limit. If $h \to 0$ from ...
1
vote
5answers
78 views

Calculate the limit $\lim_{n \to \infty}\frac{ \ln(n)^{(\ln n)}}{n!}$

I wonder what the limit $\lim_{n \to \infty}\frac{ \ln n^{\ln n}}{n!}$ would be equal to. It is well known that the factorial function grow faster than an exponential but slower than $n^n$. But how ...
1
vote
4answers
67 views

Evaluation of $\lim_{x\rightarrow 0}\left(\frac{16^x+9^x}{2}\right)^{\frac{1}{x}}$

Evaluation of $\displaystyle \lim_{x\rightarrow 0}\left(\frac{16^x+9^x}{2}\right)^{\frac{1}{x}}$ $\bf{My\; Try::}$ I am Using above question using Sandwich Theorem So Using $\bf{A.M\geq G.M\;,}$ ...
0
votes
5answers
19 views

Prove the continuity on an open interval

I need to show, that function $f(x) =\frac{2x +3}{x-2}$ is continuous on the interval $(2,\infty)$ My attempt: We should find the right-hand limit to prove the continuity: and this limit is equal to ...
0
votes
1answer
21 views

Find the radius of convergence for $\sum_{n=1}^{\infty}\frac{2^n}{3^n+4^n}z^n$

$$\sum_{n=1}^{\infty}\frac{2^n}{3^n+4^n}z^n$$ What I've done is try to evaluate the expression sans $z^n$ with the root test. $$\sqrt[n]{\frac{2^n}{3^n+4^n}}=\frac{2}{\sqrt[n]{3^n+4^n}}$$ But I'm ...
0
votes
0answers
13 views

All possible paths to evaluate a multi variable limit

Most of the books that I have (H.K Dass) say that (or at least that's what I have understood) for the limit of a multivariable function (say f(x,y) ) to exist the limit along every possible path ...
1
vote
1answer
27 views

Proof of a theorem regarding subsequences and convergence

I am attempting to understand the following theorem and it's proof as outlined in my textbook for Real Analysis. Theorem: Let $(s_n)$ be a sequence. If t is in $\mathbb{R}$, then there is a ...
1
vote
2answers
41 views

Can I subtract infinity from infinity?

I was stuck when solving a problem on limits. It was like----> $\lim_{x\to\infty} (x-x)$. What should I do now?
2
votes
3answers
50 views

How do I prove that $\lim_{z\to i} z^2=-1$?

How do I prove the following limit using the limit definition? $$\lim_{z\to i} z^2=-1$$ Using the limit definition $$|z^2+1|<\epsilon, \;\text{whenever} 0<|z-i|<\delta$$ so I factor out to ...
3
votes
2answers
34 views

$\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$

I must prove: $\lim x_n = a$, $\lim \frac{x_n}{y_n}=b$ then $\lim y_n = \frac{a}{b}$ Well, I know that $$\lim x_n = a \implies |x_n-a|<\epsilon$$ $$\lim \frac{x_n}{y_n} = b \implies ...
1
vote
1answer
22 views

$a_n, b_n$ bounded, $a_n+b_n=1$,$z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$

I must show that if $a_n, b_n$ are bounded such that $a_n+b_n=1$, and if $z_n\to a$ and $t_n\to a$, then $(a_nz_n+b_nt_n)\to a$ My idea was: $$(a_n+b_n)(z_n+t_n) = a_nz_n+a_nt_n+b_nz_n+b_nt_n$$ I ...
1
vote
3answers
53 views

$\lim a_n = L \implies \lim a_n^2 = L^2$

I have to prove the following: $$\lim a_n = L \implies \lim a_n^2 = L^2$$ I know that $\lim a_n = L \implies \forall\epsilon>0 \ \exists n_0$ such that $n>n_o\implies |a_n-L|<\epsilon$ I ...
-4
votes
0answers
18 views

How can I solve the limit with 2 variables? According to the definition of Fx, Fy [on hold]

What is the next step to solve this question (hwo i can solve it? )?CLICK HERE TO SEE THE QUESTION.
-1
votes
0answers
27 views

limit problem by 2 vars [on hold]

What is the next step to solve this question (what is the next step)? Click here to see the QUESTION
0
votes
1answer
34 views

Proving limit of the sequence ${ x }_{ n+1 }=\frac { 1 }{ 3 } \left( 2{ x }_{ n }+\frac { a }{ { x }_{ n }^{ 2 } } \right) $

Here is the question I should prove. Given: $${ x }_{ n+1 }=\frac { 1 }{ 3 } \left( 2{ x }_{ n }+\frac { a }{ { x }_{ n } ^{ 2 } } \right),{ x }_{ 0 }>0, n \epsilon \mathbb{N} $$ prove ...
0
votes
0answers
29 views

hwo i can solve the limit? [duplicate]

i have a question about the answer i got at: What is the limit of the function? someone can help me to do the next step for solve the question?
1
vote
3answers
51 views

Lebesgue Dominated Convergence Theorem example

For $x>0$ we have defined $$\Gamma(x):= \int_0^\infty t^{x-1}e^{-t}dt$$ Im trying to use Lebesgue's Dominated Convergence theorem to show $$\Gamma'(x):=lim_{h\rightarrow ...
-2
votes
0answers
73 views

hwo to solve the limit by derivative according x and y? [on hold]

hwo i can find the limit of the function the written at the picture below? $$f'_y(2,3)=-3,\quad f'_x(2,3)=2\\ \lim_{t\to0}\frac{f(2+t,3+t)-f(2,3)}{t}$$
1
vote
1answer
30 views

What is the best way to solve $\lim_{n\to \infty}{(e^{i \theta})^n}$?

What is the best way to solve the limit: $\lim_{n\to \infty}{(e^{i \theta})^n}$ $\theta$ is fixed, but you must have a care for cases $\ \theta > 0 , \ \theta = 0 , \ \theta < 0.$ There ...
-1
votes
1answer
40 views

Prove that the limit of $\frac{xy}{|y|}$ at (0,0) does not exist [on hold]

How to prove that $$\lim_{(x,y)\to(0,0)} \frac{xy}{|y|}$$ doesn't exist? I tried to prove it with paths but it's not working. Can someone help with this?
0
votes
2answers
25 views

Does $\lim\limits_{x\to a} g(x)$ exist if $\lim\limits_{x\to a} f(x)$ doesn't, and $\lim\limits_{x\to a} [f(x)+g(x)]$ does?

If $\lim\limits_{x\to a} f(x)$ doesn't exist, and $\lim\limits_{x\to a} [f(x)+g(x)]$ exists, what can be said about $\lim\limits_{x\to a} g(x)$? So I think that the answer is that $\lim\limits_{x\to ...
0
votes
2answers
28 views

$lim_{j->\infty} (j^j)/((j+1)^{j})$ [duplicate]

Can someone please explain this limit: $$lim_{j\rightarrow\infty} \frac{j^{j}}{(j+1)^{j}}=\frac{1}{e}?$$ I got it from this series: $$\sum_1^{\infty}\frac{j!}{j^j}.$$
2
votes
3answers
45 views

Rigorous Definition of One-Sided Limits

In a typical first-year Calculus course professors typically tend to put a lot of emphasis on making visual connections when working with "one-sided" limits or derivatives. This is something I find ...
0
votes
0answers
11 views

Find a limit for Doublet Stream function

In fluid Mechanics, The superimposed stream function of point source and sink is: $\psi=-\frac{Qcos\theta_1}{4\pi}+\frac{Qcos\theta_2}{4\pi}$ Graphical image of the function and for a sink - ...
3
votes
4answers
60 views

$\lim_{n \to \infty} (\frac{(n+1)(n+2)\dots(3n)}{n^{2n}})^{\frac{1}{n}}$ is equal to :

$\lim_{n \to \infty} (\frac{(n+1)(n+2)\dots(3n)}{n^{2n}})^{\frac{1}{n}}$ is equal to : $\frac{9}{e^2}$ $3 \log3−2$ $\frac{18}{e^4}$ $\frac{27}{e^2}$ My attempt : $\lim_{n ...
0
votes
0answers
37 views

A fast converging limit for $\ln x$ (or why $\ln 2 \approx \sqrt{\sqrt{42}-6})$

I've read a lot about approximating logarithms recently, and apparently it's not easy. It can be done by Taylor series (slow convergence), by continued fractions (also slow) and also by some limits. ...
-2
votes
1answer
27 views

Calculate limit of function 4 [duplicate]

Why \begin{equation} \lim_{x\to-\infty}\sqrt{x^2-x-1}-x=+\infty, \end{equation} Thanks
1
vote
0answers
47 views

How to derive this identity: $\lim _{x\to \infty} (1 + f(x))^{\frac{1}{g(x)}} = \mathrm e^{\lim_{x \to \infty} \frac{f(x)}{g(x)}}$ [on hold]

How to prove this identity: $\lim _{x\to \infty} (1 + f(x))^{\frac{1}{g(x)}} = e^{\lim_{x \to \infty} \frac{f(x)}{g(x)}}$ I've found references to this identity but no derivation This identity is ...
5
votes
2answers
179 views

Limit with x approaching infinity [on hold]

The problem says: If $$\lim_{x\to +\infty} \left\lbrack\frac{ax+1}{ax-1}\right\rbrack^x=9$$, determine $a$. It appears to be a case of $\left\lbrack\frac{\infty}{\infty}\right\rbrack^\infty$. ...
0
votes
2answers
27 views

limits of a function 3

I know these are pretty basic but i could really use your help: $\lim: \lim\limits_{x\to 0} \left(\dfrac{1+x}{ 2+x}\right)^ {(1-\sqrt{x})/ (1-x)}$ $\lim: \lim\limits_{x\to 1}\left ...
1
vote
1answer
23 views

find value or prove limit doesn't exist.

Given: Find or prove it doesn't exist: .... My attempts thus far include: I can show that doesn't exist using y=kx and showing path dependancy, but dunno if it's enough to prove that ...
-5
votes
0answers
15 views

definition of a limit in terms of sequences and epsilon delta [on hold]

link to the question i) for every sequence Xn in the real number system with limit negative infinity, we have limit f(Xn)=infinity n->infinity ii) for any M>0 and N<0 there exists a a> N such ...
1
vote
2answers
26 views

limit to infinity : trouble with l'hopital

Given the following limit for s positive constant $\lim_{x\to \infty} xe^{-sx}(\sin x-s\cos x) $ how can I prove that the above is equal to $0$ ? I re-write the limit as $ \frac{x(\sin x-s\cos ...
1
vote
1answer
31 views

Proof that: $e^{-x(1/\tau - i\xi)}$ evaluated at $x=\infty$ is zero

I remember my friend showing me how sandwich theorem can be applied here. Unfortunately, I can't find his solution anymore and I am not familiar with sandwich theory.
0
votes
3answers
46 views

Let $f : (a,b) \rightarrow R$ and $x_{0} \in (a,b)$. Assume that there are real numbers $L$ and $M$ such that … [on hold]

Could you please help me solve this problem? Thank you very much for your help. Let $f : (a,b) \rightarrow R$ and $x_{0} \in (a,b)$. Assume that there are real numbers $L$ and $M$ such that ...
1
vote
1answer
30 views

The function $\lim_{n\to\infty}({4^n+x^{2n}+\frac{1}{x^{2n}})}^{1/n}$ is non-derivable at

The function $$f(x)=\lim_{n\to\infty}\left(4^n+x^{2n}+\frac{1}{x^{2n}}\right)^{\frac{1}{n}}$$ is non-derivable at how many points? The limit is of $\infty^0$ form. Is it an indeterminate form or ...
1
vote
1answer
14 views

Why does the average of a set of random numbers to the nth power approach 1/(n+1)?

I got bored and started running a Java program to mess with stuff like this. I did a boatload of trials and averaged them all together, first for a random number squared. Quick pseudo-code: ...
-1
votes
2answers
71 views

What is the $\lim _{x→\infty}(0.5)^x$? [on hold]

I need to verify the answers to two questions: 1)What is the $\lim_{x→∞}$ $(0.5)^x$? A: $.25$ 2)What is the value of $\lim_{h→0}$ $(8^h−1)/h$ A: $-1$ Are my answers correct? If not, what are ...
1
vote
1answer
40 views

Show the Cauchy-Riemann equations hold but f is not differentiable

Let $$f(z)={x^{4/3} y^{5/3}+i\,x^{5/3}y^{4/3}\over x^2+y^2}\text{ if }z\neq0 \text{, and }f(0)=0$$ Show that the Cauchy-Riemann equations hold at $z=0$ but $f$ is not differentiable at $z=0$ ...
2
votes
4answers
30 views

Limit of a complex sequence

So I wanted to calculate $$\lim_{n\rightarrow\infty}\frac{n^2}{(4+5i)n^2+(3+i)^n}$$ I thought that I could do it easier if I calculate $\lim_{n\rightarrow\infty}\frac{(3+i)^n}{n^2}$. First I write ...
0
votes
0answers
11 views

Limit over general chain

We all know classic definition of limit of a sequence. There is also definition of limit of a function. Now consider general chain, i.e. linearly ordered set $(\mathcal{I}, \le)$ with specified values ...
-1
votes
1answer
24 views

Supremum calcualtion, how do I get the answer [on hold]

Find the supremum of G={x|(x^2+1)^−1>1/2} Could someone help me out with this calculation? Thanks. Appreciate. Options are: ...
1
vote
0answers
30 views

The space of sequences which are eventually zero in $l^2$ is not a Hilbert space.

Define $V$ to be the space of sequences which are eventually zero, i.e. $$V=\bigcup_{N=1}^{\infty}\{(x_n)_{n=1}^{\infty}\in l^2: x_n=0 \; \text{for}\; n\ge N\}.$$ Is $V$ a Hilbert space with ...
1
vote
3answers
48 views

$\lim_{x\rightarrow\infty}e^{-x}\cosh(\alpha x)$

How can I compute this limit with $\alpha>0$. $$\lim_{x \rightarrow \infty} e^{-x} \cosh(\alpha x)$$ For $\alpha=1$ it is simple, but if $\alpha \neq1$ it isn't simple. Thank you very much.
-1
votes
1answer
46 views

If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? [on hold]

If $\lim_{x\rightarrow\infty}f(x)=0$, does $||f||_{L^2}=0$? Thank you very much.
0
votes
0answers
23 views

Existence of limit and continuity for a function defined at discrete points

Consider the function $f(x)= \arcsin (\frac{1+x^2}{2x})$. Due to the following two inequalities: (i) $1+x^2 \geq 2x$ (ii)$1+x^2 \geq -2x$ , the function is only defined at $x=1$ and $x=-1$. Is it ...
0
votes
1answer
53 views

Why doesn't L'Hospital's rule work for this limit?

Let $a,b,A,B$ be positive real numbers, with $a>b$ and $A>B$. Consider the limit: $$\lim_{x\to\infty}\frac{ax+b\sin(x)}{Ax+B\sin(x)}$$ By the squeeze theorem, the limit exists and is equal to ...
-2
votes
1answer
47 views

Limit evaluate calculate [on hold]

How to (correctly) evaluate this limit? $$\lim_{x\to0}\left(\frac{c}{e^{1/x^2}x^j}\right) = 0$$ when $j \ge 1$ and $c$ is a constant for any $x$.
1
vote
4answers
67 views

How to evaluate $\lim _{x\to \infty }x\left(\arctan x-\frac{\pi }{2}e^{1/x}\right)$?

I have a problem with this limit, I have no idea how to compute it. Can you explain the method and the steps used (without L'Hôpital if is possible)? Thanks $$\lim _{x\to \infty }x\left(\arctan ...