# Tag Info

1 vote

### Limit of $f(x,y)$ when $x^2+y^2$ tends to $\infty$

obvious ho to get large positive values, say $x=y$ or $x=7y.$ Next, given an $x,y$ pair you like, what is $u^2 + 3uv + v^2$ when $$u = 5x + 2y, \; \; \; v = -2 x - y \; \; \; ? \; \;$$

1 vote

### Doubts: continuity VS existence of the limit

Generally speaking, a textbook about continuous real functions will proceed in this order: First, define continuity at a point contained in the domain of the function. Then, define left-continuity at ...

Accepted

### Doubts: continuity VS existence of the limit

The domain of the square root is $[0,∞)$, so you cannot approach $0$ from below. It therefore suffices to approach $0$ from above! The limit should be the same no matter how one approaches $0$, and in ...
1 vote

### Solutions to $x^x=1$?

Your question is $x^{x}=1$. Now clearly $x=1$ satisfies the above equation because $1^{1}=1$. Again you can solve this above equation by taking $log$ on both sides. $x^{x}=1$. Therefore $xln(x)=ln(1)$....

### Proof of Convergence: Babylonian Method $x_{n+1}=\frac{1}{2}(x_n + \frac{a}{x_n})$

An easy way to prove the Babylonian method works is to think of it as like binary search, but faster. With each $x_n$, associate the closed interval $I_n$ between $x_n$ and $\frac {a}{x_n}$, inclusive....
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### On kernels and stalks of sheaves.

For 1., there is indeed a general approach: given a category $\mathcal{C}$ and a (full) reflective subcategory $\mathcal{D}$ of it (i.e. the inclusion $i\colon \mathcal{D}\to\mathcal{C}$ is fully ...

### Find $\lim_{n\to\infty}\left(\int _{0}^{n}\frac{\mathrm dx}{x^{x^n}}\right)^n$

Let $f_n(x)=x^n\log x$. It is not difficult to show that $$I_n=\int^n_0e^{-f_n}\xrightarrow{n\rightarrow\infty}1$$ This may be achieved by analyzing the integral over $(0,1]$ and $(1,\infty)$ ...
1 vote

### Compute $\lim_{n\to\infty}(\frac{a_{n+1}}{\sqrt[n+1]{b_{n+1}}}-\frac{a_n}{\sqrt[n]{b_n}})$ given $(a_{n+1}-a_n)/n\to a$ and $b_{n+1}/(nb_n)\to b$

Here is a rather nice result which helps in this kinds of problems: Theorem (O. Carja): Suppose $(a_n:n\in\mathbb{N})$ is a sequence of positive numbers such that (i) $\lim_n\frac{p_n}{n}=p>0$ (...

### Computation of limit involving specific differences of logarithms

Hint $$\sum_{k=a}^b \log(k)=\log \Bigg(\prod_{k=a}^b k\Bigg)=\log \Bigg(\prod_{k=1}^b k\Bigg)-\log \Bigg(\prod_{k=1}^a k\Bigg)= ???$$