# Tagged Questions

For question about limit theorems of probability theory, like the law of large numbers, central limit theorem or the law of iterated logarithm.

27 views

### Asymptotic Relative Efficiency: Poisson

I'm trying to find the asymptotic relative efficiency of a Poisson process: $\frac{\lambda^t exp(-\lambda)}{t!}$ = P(X=t). When X = t = 0, the best unbiased estimator of $e^{-\lambda}$ is ...
29 views

### Compute imaginary part of the limit

I am trying to compute the imaginary part of this particular question $$\lim_{z\rightarrow i} \frac{iz^3}{z+9 i}$$
18 views

### Find $\alpha$ such that $\lim_{(x, y) \to (0, 0)} \frac{|x|^\alpha y}{x^2 + y^2} = 0$. Can I simply move to polar coordinates?

This is the full exercise: Given function: $$f(x, y) = \begin{cases} \frac{|x|^\alpha y}{x^2 + y^2}, & \text{if }(x,y) \ne (0, 0) \\ 0, & \text{if } (x,y) = (0,0) \\ \end{cases}$$ ...
23 views

### Is $\lim_{n\rightarrow\infty }nz^{n!n}=0$ for $|z|<1$?

Is $\lim_{n\rightarrow\infty }nz^{n!n}=0$ for $|z|<1$? We have a $\infty \cdot 0$ case, then how we proceed? How to use the L'Hospital's Rule? Thanks in advance!
48 views

### determine the limit of sequence?

Need help on how to determine the limits, if it exists for this sequence. Have no idea where to start. Thanks in advance smart people! $$A_n = \dfrac {(n+2)^{2n}}{(n^2-n-6)^n}$$
30 views

### how to find existence and value of limit in multivariable calculus

I was in maths class and i found a question interesting. Find the limit of $\lim_{(x,y)\to (0,0)} \frac{2x}{x^2+x+y^2}$ if it exist.one of my friend did this question by transforming into polar ...
27 views

### Simple Limit Question with Denom 0

I was asked to find the limit of the following: $$\lim_{x\to -1}\frac{\sqrt {x ^ 2 + 8} - 3}{x + 1}$$ I have tried using the Limit Laws but am always getting $\frac {0}{0}$. The answer given is ...
52 views

46 views

### Confusion with this definition of the derivative

This function is from my text: $$p(\theta) = \sqrt{13\theta}$$ It states that the derivative of the function $p(\theta)$ with respect to the variable $\theta$ is the function $p'$ whose value at ...
31 views

### Clarifying a Step in Central Limit Theorem Derivation

So I'm trying to understand how to derive the central limit theorem but I'm confused about step 3 in Peter Young's Derivation In step 3, rightmost part of the equation, why are we taking the sum ...
35 views

### Prove $\limsup X_n = 1$ has probability 0

If $X_n$ are i.i.d. random variables U[0,1], is it true that $$\{\omega : \limsup X_n(\omega) =1\}$$ has probability 0? How would you prove that?
107 views

### Limits and continuity: a real analysis proof.

Let $E \subseteq R$, $p$ is limit point of $E$ , and $f\colon E\to R$. Suppose there exist a constant $M>0$ and $L\subseteq R$ such that $|f(x)-L| \leq M|x-p|$ for all $x\subseteq E$. Prove ...
52 views

### proof of differentiatiable function

prove that x^(1/3) is differentiable at a with f(a)'=((a^(1/3))^-2)/3 for all a not equal to 0. I tried a epsilon-delta proof with limes theorem, and or that does not work or I am making somewhere ...
106 views

### prove if $(A_n)$ limit is $L$ then $(A_n)^2$ limit is $L^2$

Hello, What I need to prove is: if $(A_n)$ limit is $L$ then $(A_n)^2$ limit is $L^2$. I've added my attempt to prove it. I got stuck so I'm guessing I'm missing something here. help will be ...
160 views

### How to evaluate this limit of irrational function?

$$\lim_{h \rightarrow 0}\frac{5}{\sqrt{5h+1}+1}$$ Some things I'm confused about: 1) Why should we rationalize the denominator? It won't get rid of the square root, it will just move it to the ...
50 views

### Limit of Identity Function vs. limit of Squaring Function

$$\lim_{x\rightarrow a} x = a$$ and $$\lim_{x\rightarrow a} x^2 = a^2$$ $f(x)=x^2=x \times x$, i.e.: two identity functions. I'm a bit confused on how $x^2$ can be interpreted as being ...
87 views

### Central Limit Theorem Clarification

The Central Limit Theorem states that the sampling distribution of the sample mean: Converges in distribution to a normal distribution. Has an expected value (mean of the sampling distribution of ...
89 views

### $\lim_{x\to \infty}\dfrac{x^n}{x+e^x}$ without using LHR

Prove that for each n∈N. $\lim_{x\to \infty} \dfrac {x^n}{x+e^x}=0$ using basic $\varepsilon$ - $N$ method. I tried as We have $\left|\dfrac{x^n}{x+e^x}\right|<\varepsilon$. I take $x>1$ and it ...
42 views

77 views

### Differentiability Theorem Question

$f(x,y) = \begin{cases} \frac{1}{2} y \log(x^2+y^2), & (x,y) \neq (0,0) \\ 0, & (x,y) = (0,0) \end{cases}$ You may assume that this is a continuous function. Prove that f does not satisfy ...
### How to compute $\lim_{n\to\infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}3\right)^n$?
How would one compute $$\lim_{n\to\infty}\left(\frac{a^{1/n}+b^{1/n}+c^{1/n}}3\right)^n$$ if $a,b,c>0$. I've never done an integral 3 variables before! This is from a chapter called Interchange of ...
### Show the the series $\sum_{n,m=1}^\infty 1/(n+m)!$ is absolutely convergent and find its sum.
Show the the series $$\sum_{n,m=1}^\infty \dfrac{1}{(n+m)!}$$ is absolutely convergent and find its sum. This comes from a chapter called interchange of limit operations. I tried using the ratio ...