Articles of limits

How do I prove that $\lim_{(x,y) \to (0,0)} \frac{2y^2}{\sqrt{x^2+xy}}$ exists?

Now I have learnt that to prove a function of 2 variables exists we must have the both the repeated limits as equal, which is $\lim_{(x=0,y\to0)}f(x,y) = \lim_{(x\to0,y=0)}f(x,y)$ , now in this case $f(x,y)= \frac{2y^2}{\sqrt{x^2+xy}}$ so $\lim_{(x=0,y\to0)}f(x,y) = \lim_{(x=0,y\to0)} 2y^2/\infty$ which is not defined ,while for $\lim_{(x\to0,y=0)}f(x,y) = 0$ which is I think is enough […]

Computing a Double Limit

How would one compute $\lim_{\delta \rightarrow 0, k\rightarrow\infty} (1+\delta)^{ak}$, where $a$ is some positive constant? I am finding a lower-bound of the Hausdorff Dimension on a Cantor-like set and this expression appeared in my formula. Here’s what I have, even though I’m not sure if I can use L’Hopital in this case (where $k, \delta$ […]

Find $\lim _{x\to \infty }\left(x\left(\ln\left(x+1\right)-\ln x\right)\right) $

Find $$\lim _{x\to \infty }x\Big(\ln\left(x+1\right)-\ln x\Big)$$ Here’s how I do it: $$x\Big(\ln(x+1)-\ln x\Big) = x\Bigg(\ln(x(1+\frac{1}{x}) – \ln x\Bigg)$$ $$x\Big(\ln x + \ln(1+\frac{1}{x}\Big)-\ln x =x\ln\Big(1+\frac{1}{x}\Big)$$ $$\ln\Bigg( \Big(1+\frac{1}{x}\Big)^x \Bigg) \rightarrow \ln 1^\infty = 0$$ What am I doing wrong? The answer is supposed to be $1$, but I get $0$.

What's $\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$?

This originally comes from $f_1(x,y)=\frac{x}{y}$, where $X=\mathbb{R}^{n}, Y=\mathbb{R}^m, x \in X, f: X \rightarrow Y, x \neq 0, f(x) \neq 0$ $$\limsup_{(h_x,h_y)\to(0,0)} \frac{\left|\frac{x+h_x}{y+h_y}-\frac{x}y\right|} {\sqrt{{h_x}^2+{h_y}^2}}$$ If I understand it correctly, $x$ and $y$ can be anything but zero, and $h_x,h_y$ go towards zero. Moreover, both numerator and denominator cannot be negative. But since $x,y$ could be […]

How to find this limit with following constraints?

Question: If a,b are 2 positive , co-prime integers such that $$\lim _{n \rightarrow \infty}(\frac{^{3n}C_n}{^{2n}C_n})^\frac{1}{n}=\frac{a}{b}$$Then a+b?? I tried to break down the limit to: $$\frac{[(3n)(3n-1)\dot{}\dot{}\dot{})^\frac{1}{n}][n(n-1)(n-2)\dot{}\dot{}\dot{})^\frac{1}{n}]}{[2n(2n-1)(2n-2)\dot{}\dot{}\dot{}]^\frac{2}{n}}$$ but I’m lost ahead of it. Answer given in my text books is 43. Please guide me

Two simple series

I dont know how to calculate these two series: $$\begin{align} & \sum\limits_{n=1}^{\infty }{\frac{n+3}{{{n}^{3}}+\ln n}} \\ & \sum\limits_{n=1}^{\infty }{\left( 1-\cos \frac{\pi }{n} \right)} \\ \end{align}$$

How to integrate this type of integral: $\lim_{n \rightarrow \infty}{\int_ 0^1}\frac{n x^{n-1}}{1+x}dx$

This question already has an answer here: Limit of $s_n = \int\limits_0^1 \frac{nx^{n-1}}{1+x} dx$ as $n \to \infty$ 5 answers

An apparently new method to compute the $n$th root of any complex number

I found  a series of articles (in Portuguese) by a Brazilian mathematician named Ludenir Santos, where presents a series of iterative methods, he said new, to extract nth roots of any complex number different from zero (and hence for real numbers), with any desired degree of accuracy.  In his first articles, he has two different methods […]

The limit of hazard rate $h(x)=A/(1-B)$ as $x$ approaches $\pm \infty$

Can we tell what happens to the limit as $x$ approaches $\pm \infty$ of a hazard rate $h(x)$ defined for unspecified or generalized density as: $$ h(x)=A/(1-B) $$ where $A=f(x)$ is the density function, and $B=F(x)$ is the CDF.

The link between the monotony of a function, and its limit

Let’s assume I have a convergent function f, as x approaches to $$+\infty$$. Is-it true to say that it exists a real x0, such that forall x>x0, f is either increasing or constant or decreasing ? (And if it is true, how do I prove it ? By using the convergent in +infty definition ?) […]