# All Questions

2 views

### Solving limit without L'Hôpital

I need to solve this limit without L'Hôpital's rule. These questions always seem to have some algebraic trick which I just can't see this time. $$\lim_{x\to0} \frac{5-\sqrt{x+25}}{x}$$ Could ...
38 views

### single simple pole, $e^{i\theta } \frac{1-\overline{z_0}z}{z-z_0}$

$D= \big\{z:|z|\leq 1\big\}$, $|z_{0}|\in D$. A function $f(z)$ such that 1).$f(z)$ is analytic on $D\setminus \{z_0\}$; 2).$f$ has a single simple pole at $z_0$ ; 3). $f(z)\ne 0$, ...
15 views

### Logic type problem.

A school has 90 children.During the day each child attends 4 classes.Each class has 15 children and 1 teacher.During the day each teacher has 3 classes.What is the smalles number of teachers the ...
80 views

9 views

### The boundary area of union of balls

Let {v_i} be countable many points in an n-dimensional unit cube. Let A be the union of all of the balls around {v_i} which have radius 1/10. Is it true that the (n-1)-dimensional Hausdorff measure of ...
14 views

### Does a differentiable function map open intervals to open intervals?

I know that preimages are open for open images, since differentiability implies continuity. I suspect there is a counterexample to the above though. This is not homework, just study.
4 views

### The natural projection mapping $\pi : G -> G/N$ defined by $\pi(x) = xN$, for all $x$ in $G$, is a homomorphism, and $\ker(\pi) = N$.

The question again... The natural projection mapping $\pi : G -> G/N$ defined by $\pi(x) = xN$, for all $x$ in $G$, is a homomorphism, and $\ker(\pi) = N$. I am wanting to prove that ...
41 views

### How do I evaluate this integral by hand?

TL;DR how do I evaluate $\int_0^{2 \pi } \frac{1}{\cos ^2(\theta )+1} \, d\theta$ by hand? I'm trying to solve this problem: Find the volume of the region defined by $x^2+xy+y^2+yz+z^2\le1$. ...
10 views

8 views

### Riemman sum to get surface area of cone, and cone height

I have this itch to think about math ideas in my free time. Like when I play a game or something, I kind of "work in parallel" on math ideas. Back in one of my calculus classes we worked on Riemann ...