Questions on the evaluation and properties of limits.

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0
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1answer
10 views

Limit $n \rightarrow \infty \frac{n}{e^x-1} \sin\frac{x}{n}$

I am just working through some practice questions and cannot seem to get this one. Plugging this into wolfram alpha I know the limit should be $\frac{x}{e^x-1}$, but I am having a bit of trouble ...
0
votes
1answer
54 views

what's the limit $\lim\limits_{n \to \infty} \sum_{k=n}^{\infty} e^{-k^2}$

I have no idea how to compute the tail sum $\lim\limits_{n \to \infty} \sum_{k=n}^{\infty} e^{-k^2} $. I tried subtracting the first n items from all but realized that I don't know a way to calculate ...
1
vote
1answer
22 views

L'Hopital's rule and limiting variables

I'm working some problems from a calculus text and came across this question: If $f(x)$ is a function that's differentiable everywhere, what is the value of the limit $$\lim\limits_{h \to ...
4
votes
1answer
41 views

Find the limit analytically when the sine functions have square roots?

Find the limit analytically of the following: $\lim \limits_{x \to 0} \frac {\sin(\sqrt{2x})}{\sin(\sqrt{5x})} $ The closes thing we learned in class about this was that $\sin(x)$ over $x$ will ...
1
vote
1answer
20 views

Convergence Proof Help?

Let ($x_n$) and ($y_n$) be given, and define ($z_n$) to be the "shuffled" sequence ($x_1, y_1, x_2, y_2, x_3, y_3,...x_n, y_n$). Prove that ($z_n$) is convergent if and only if ($x_n$) and ($y_n$) are ...
2
votes
1answer
31 views

Proving that limit of a sequence is 0 from definitions.

I had this question in a test: Use the definition of limit in order to prove that if $\{a_n\}$ (n goes from 1 to infinity) is a sequence of real numbers such that $\lim_{n\rightarrow \infty} a_n^2 ...
0
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2answers
31 views

Proving $\lim_{n \to \infty} 10000 (\log n)^{1000}/(n^{1.0001}/1000) = 0$

Initially it seems that $10000(\log n)^{1000}$ is far greater than $n^{1.0001}/1000$, but Wolframalpha says that $$\lim_{n \to \infty} \frac{10000 (\log n)^{1000}}{1.0001 n^{1.0001}/1000} = 0$$ I ...
0
votes
1answer
18 views

Showing a multivariable function isn't continuous.

Suppose I wanted to show some multivariable (specifically, 2 variables, is what im referring to) function wasn't continuous. What ways are there to go about doing that? From what I know, there seem to ...
0
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0answers
27 views

Epsilon-Delta Limit for Trigonometric Function

I'm studying an Epsilon-Delta proof for a trigonometric function: $$\lim_{x \to 1/9} \sin(x) = \sin(1/9)$$ This is the procedure from my (Italian) book: $$−\epsilon < \sin(x) − \sin(1/9) < ...
1
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2answers
53 views

Generalizing limits of sums, products, and quotients of sequences to abstract topological spaces?

Introductory real analysis books usually state some properties about limits of sequences of numbers. Suppose $\lim x_n=x$ and $\lim y_n=y$. Then: $\lim (x_n+y_n) = x+y$ $\lim c x_n = cx$, for $c \in ...
0
votes
2answers
32 views

What is the meaning of Right Hand Limit at $\infty$?

For a limit to exist, the left hand limit must equal the right hand limit. That is, $$\lim_{x\to c^+}f(x)=\lim_{x\to c^-} f(x)$$ However, if $x\to\infty$, then what does the right hand limit mean ? ...
0
votes
5answers
37 views

Solving a limit

Question: $$\lim_{x \to \infty} \bigg(\frac{x+6}{x+1}\bigg)^{x-4}$$ Attempt: It is quite obvious that the term inside the brackets tends to $1$ while the exponent tends to $\infty$. How would I ...
1
vote
1answer
47 views

Limit of Multi-variable Function

Question What condition must non-negative integers m, n and p satisfy so that $$\lim_{(x,y)\to(0,0)}\frac{x^my^n}{(x^2+y^2)^p}$$ exist? Prove your answer. [Note: if $m=n=p=0$, then the limit ...
3
votes
3answers
80 views

Why does $e^{-x}$ approach $0$ as $x$ gets large? [on hold]

Why is it that $$\lim_{x \to -\infty} e^x = 0?$$
4
votes
2answers
51 views

Limiting value of $\frac{x^n e^x}{n!}$ as $n\to\infty$

For the Taylor Series the remainder is of the form $$R_n = \frac{(x-a)^n}{n!} f^{(n)}(\xi) $$ with $a \leq \xi \leq x$ For the series of $e^x$ about $0$ (that is, the Maclaurin series) the remainder ...
3
votes
2answers
71 views

Problem with two variable limit where $\lim\limits_{(x,y) \to (-1, 8)} xy = -8$ using only definition

So we just started with two-variable limits. The definition is quite straight forward though my head is still giving it a few spins. I thought doing a couple of examples would help me. Come the second ...
2
votes
6answers
57 views

Limit of $\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$

Limit of $$\lim_{x \rightarrow 0} \frac{\sin xy^2}{x}$$ I know (thanks to wolfram) it is equal to $y^2$, but i do not know how to show that.
0
votes
2answers
39 views

Find the limit of $\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$

Find the limit of: $$\lim_{(x,y)\rightarrow(+\infty, +\infty)}\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)}$$ I think the solution could be: $$\frac{x+y+\sin xy}{x^2+y^2+\sin^2 (xy)} \le \frac{x+y+\sin ...
3
votes
3answers
101 views

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$?

How to find $\lim_{n\to\infty}\left(\frac{\pi^2}{6}-\sum_{k=1}^n\frac{1}{k^2}\right)n$? It is well-known that $\lim_{n\to\infty}\sum_{k=1}^n\frac{1}{k^2}=\frac{\pi^2}{6}$, so ...
-1
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2answers
29 views

CPT exam Quantitative aptitude exercise 8c [on hold]

$$\lim\limits_{n\to \infty}\left[\frac{1}{6}+\frac{1}{6^2}+\frac{1}{6^3}+\cdots+\frac{1}{6^n}\right]$$ is: (a) $\frac15$ (b)$\frac16$ (c)$-\frac{1}{5}$ (d) none of these According to book ...
0
votes
1answer
18 views

Limit of a Monotonic Increasing and Non-Bounded Function

I have made a solution for the following question and I'm wondering if it's correct. I think that something is missing here. Can you help me complete the solution? Let $f$ be a function. The ...
1
vote
0answers
63 views

Limit of recursive sequence $x_{n+1}=\frac1n(x_1+2x_2+3x_3+…+nx_n)$

I was trying to solve the following limit but I just can't get it: Let $x_1 = a$, $a>0$, and, for every $n \in \mathbb{N}$, $$x_{n+1}=\frac{x_1+2x_2+3x_3+...+nx_n}{n}.$$ Determine : ...
1
vote
1answer
44 views

How to prove the following limits [on hold]

$$\mathop {\lim }\limits_{x \to 1} \left\{ { - \sum\limits_{n = 1}^\infty {\frac{{{x^{2n}}}}{{{{\left( {n + a} \right)}^s}}}} \ln \left( {1 - x} \right) - \sum\limits_{n = 1}^\infty ...
1
vote
1answer
59 views

What is the limit of the sum of “last half” part of harmonic series?

I'm looking for the limit of this sum: $\frac{1}{\left\lceil\frac{n}{2}\right\rceil+1}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+2}+\frac{1}{\left\lceil\frac{n}{2}\right\rceil+3}+\cdots+\frac{1}{n}$ ...
0
votes
3answers
50 views

Solving for a Limit Given a Limit

$$ \text{Given}\; \lim_{x \to 1} \frac{f(x)-4}{x-1} = 10, \;\text{evaluate}\; \lim_{x \to 1} f(x) $$ I'm wondering if anyone can give me some tips on how to approach this problem. I ...
2
votes
1answer
31 views

Limit of a recurrence

I was given the following exercise as homework: find the limit of $b_{n+1} = \sqrt{2 + b_n}$, $b_1 = \sqrt{2}$, with a hint that $b_n < 2 \forall n \in \mathbb{N}$. I have proven that $b_n$ is ...
1
vote
2answers
85 views

What's the answer to this limit question?

Can anyone find the limit to this one? $\lim_{n \rightarrow \infty} (\sum_{i=(n+1)/2}^n {n \choose i} \times 0.51^i \times 0.49^{n-i})$ When I plot it, it seems to me to approach 1, which makes me ...
4
votes
5answers
80 views

Does the limit $\lim\limits_{x\to0}\left(\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}\right)$ exist?

Does the limit: $$\lim\limits_{x\to0}\frac{1}{x\tan^{-1}x}-\frac{1}{x^2}$$ exist?
1
vote
1answer
26 views

Limit Help: $\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}$

I feel dumb for asking this, but I couldn't quite show that this limit is 0 (which I think is correct) whenever $a>0$: $$\lim_{x\to\infty} xe^{-a\frac{x}{\ln x}}.$$ I tried using L'Hospital's ...
1
vote
1answer
34 views

Computing limits example: Swaping limit to $0$ into infinity.

I have found the following example: $$ \lim_{x\to 0^{+}} \frac{e^{\frac{-1}{h}}}{h} = \lim_{z\to\infty} ze^{-z} = 0 $$ Could you explain to a kid nice and slowly why does the limit of $x$ to ...
11
votes
1answer
63 views

Show that $\lim\limits_{n\to\infty}\frac1{n}\sum\limits_{k=1}^{\infty}\left\lfloor\frac{n}{3^k}\right\rfloor=\frac{1}{2}$

Show that $$\lim_{n\to\infty}\frac1n\sum_{k=1}^{\infty}\left\lfloor\dfrac{n}{3^k}\right\rfloor=\frac{1}{2}$$ I can do right hand. $$\sum_{k=1}^{\infty}\left\lfloor\dfrac{n}{3^k}\right\rfloor\le ...
-3
votes
0answers
42 views

Prove the limit of complex function can be broken into real and imaginary parts [on hold]

let $f(z)$ be a function of a complex variable and let $a,l \in \mathbb C$. Prove that $\lim\limits_{z \to a} f(z)=l \Longleftrightarrow \lim\limits_{z \to a} \operatorname{Re} f(z)=\operatorname{Re} ...
0
votes
0answers
31 views

Differentiating CDF

I'm trying to differentiate the cdf of z with respect to x where the upper bound is a function of x and z ~ N(a , $b^2$ $\cdot$ $x^{-2}$) $\frac{d}{dx} \int _{-\infty} ^ {z^*(x)} \Phi ^{\prime} (z) ...
0
votes
3answers
36 views

If limit of $ \lim_{x\to0}(\frac{sin2x}{x^3} + \frac{a}{x^2} + b) $ is zero, then find a+b? [on hold]

If limit is zero: $$ \lim_{x\to0}\left(\frac{\sin 2x}{x^3} + \frac{a}{x^2} + b\right) = 0 $$ then find $ a+b=? $ please help me to solve this question, thanks.
0
votes
2answers
48 views

How to calculate the limit of $\frac {2x^2+x-1}{x^2}$?

How do I calculate the limit of $\frac {2x^2+x-1}{x^2}$ when $x$ approaches $0$? What I do is I immediately check what happens when $x$ approaches $0+$ and when $x$ approaches $0-$ and then on both ...
2
votes
6answers
116 views

Find the limit of the sequence $(4^n)/((2n)!)$

How to find $$\displaystyle \lim_{n\to\infty} \frac{4^n}{(2n)!}$$ Can I use l'Hopital's rule?
0
votes
1answer
25 views

Choose the value of k that makes the following function continuous at x = -6

$$\begin{cases} kx + 8 & x < -6\\ -9x + k & x \geq -6 \end{cases}$$ When I did my work $$kx+8 = -9x+k\\ k(-6)+8 = -9(-6)+k\\ k(-6)+8 = 54+k\\ k(-6) = 46+k$$ How do I go from here?
0
votes
1answer
32 views

What is the big Oh notation for the following series.

I have the series $1+3+9+27+... + 3^n$ . I need to find the Big O solution. What I have tried. The above series is a Geometric Progression with r=3. SO the sum would be. $ [1* 3^{n+1} - 1]/2 $ How ...
1
vote
3answers
73 views

How to calculate the limit of $(\frac{x}{x+1})^x$

I am looking at the probability of losing $x$ games in a row, in a game where the probability of winning is $1/x$. (For example, if this is a fair casino game, what is the probability of losing $x$ ...
-2
votes
3answers
41 views

$\lim_{x\to \frac54\pi} \lfloor\sin x+ \cos x\rfloor$ where $\lfloor\cdot\rfloor$ is Greatest Integer Function, is $-1$ or $-2$? why?

There's a few questions in limits that I think I got right but my peers have challenged my answers and I have no idea how to solve these limits to get the answers they're getting or if I'm right!! ...
2
votes
1answer
36 views

Doubt in raising a power to a complex number

What's the value of $$i^{i^{i^{...}}}$$? I tried to take log on both sides. $x=i^x$ $\implies \log x=x \log i$ After this how can I solve this... I am sorry, that I don't know the methods you ...
0
votes
1answer
14 views

verify indicated limit using formal definition

I have this exercise "Use the formal definition of limit to verify the indicated limit." $ \lim_{x \rightarrow 1} (3x + 1) = 4 $ ... I have made an attempt but Im kinda stuck. So I guess it goes ...
0
votes
1answer
43 views

Limits in multivariable function

$$\lim \limits_{(x, y) \to (0,0)} {x^3 + \sin(x^2+y^2)\over{y^4 + \sin(x^2+y^2)}}$$ I don't visualize a limited function anywhere to evaluate this limit (by the way, I have the information that this ...
0
votes
1answer
32 views

How do I solve this limit

**Find the limit ${x \to 6}$ if $ f(f(x))$ ** Any idea how to solve this graphical question on limits?
1
vote
1answer
24 views

$\lim_{n\to\infty} d^{-n}e^{o(n)}, d>1$

I think it's rather a silly question, but I have some problems to say definitely what $$ \lim_{n\to\infty}d^{-n}e^{o(n)} $$ is, where $d>1$. Of course, $d^{-n}\to 0$ as $n\to\infty$. Is it true ...
0
votes
0answers
15 views

Proving that a random variable is stable

If $\overline{F} = 1- F_X$ where $F_X$ is a cumulative distribution function of $X$. Then if the following is satisfied: $$ \lim_{x \to \infty} \frac{\overline{F}(\lambda x)}{\overline{F}( x)}= ...
1
vote
3answers
54 views

Limits using definite integration

$F(k)$ = $$ \lim_{n\to \infty}{\frac{1^k + 2^k +...+n^k}{(1^2 + 2^2 +...+n^2)*(1^3 + 2^3 +...+n^3)}} $$ I need help in finding $F(5)$ and $F(6)$. I tried converting it into summation form and using ...
0
votes
1answer
13 views

Limit with épsilon and delta , multivalues

Hi i need help with about the how to apply the definition of limit correctly, the following problem $ \lim \limits_{(x,y)\to (5,6)}x^{2}+6y^{2}-7$ I appreciate your sugerences.
2
votes
2answers
22 views

Finding Tangent Line Using Limit Definition

I'm supposed to get the equation of the tangent line to the graph of $f(x)= \frac{8}{x}$ at the point $(2,4)$. I started with $$\frac{\frac{8}{x+h} - \frac{8}{x}}{h},$$ then I cross multiplied: ...
0
votes
0answers
29 views

Solving for the limit of a Gaussian random variable within an integral

I'm having trouble solving a particular integral. It is $$ (1/\Delta t)\int_t^{t+\Delta t}I(t')dt', $$ where $$ I(t') = \mu_c+\sigma_c \eta(t'). $$ In this second equation, $$ \eta(t') = ...