Articles of radicals

Prove $\lim\limits_{n \to \infty} \sqrt{X_n} = \sqrt{\lim\limits_{n \to \infty} X_n}$, where $\{X_n\}_{n=1}^\infty$ converges

Let $\{X_n\}_{n=1}^\infty$ be a convergence sequence such that $X_n \geq 0$ and $k \in \mathbb{N}$. Then $$ \lim_{n \to \infty} \sqrt[k]{X_n} = \sqrt[k]{\lim_{n \to \infty} X_n}. $$ Can someone help me figure out how to prove this?

Proving the irrationality of $\sqrt{5}$: if $5$ divides $x^2$, then $5$ divides $x$

I am working on proving that $\sqrt{5}$ is irrational. I think I have the proof down, there is just one part I am stuck on. How do I prove that $x^2$ is divisible by 5 then $x$ is also divisible by $5$? Right now I have $5y^2 = x^2$ I am doing a proof by […]

Find minimal value $ \sqrt {{x}^{2}-5\,x+25}+\sqrt {{x}^{2}-12\,\sqrt {3}x+144}$ without derivatives.

Find minimal value of $ \sqrt {{x}^{2}-5\,x+25}+\sqrt {{x}^{2}-12\,\sqrt {3}x+144}$ without using the derivatives and without the formula for the distance between two points. By using the derivatives I have found that the minimal value is $13$ at $$ x=\frac{40}{23}(12-5\sqrt{3}).$$

Prove $\lim_{x\to0}\sqrt{4-x}=2$ using the precise definition of limits. (Epsilon-Delta)

Prove $$\lim_{x\to0}\sqrt{4-x}=2$$ using the precise definition of limits. (Epsilon-Delta) I am not sure how to link $0<\left |x \right |<\delta$ with $\left |\sqrt{4-x}-2\right |<\epsilon$ . EDIT (Trying it out now) I worked till here, then I basically got stuck.

Find $\,\lim_{x\to 1}\frac{\sqrt{x}-1}{\sqrt{x}-1}$

How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$. Please help me. Thanks!

Limit of $\sqrt{x^2-6x+7}-x$ as x approaches negative infinity

What is $\lim\limits_{x\to-\infty}(\sqrt{x^2-6x+7}-x)$ ? Don’t understand how to approach this question

$f(n) = \int_{1}^{n} n^{x^{-1}}dx$. $\frac{df}{dn}$?

Lets start with the domain $1 \leq n \in \mathbb{R}$: $f(n) = \int_{1}^{n} n^{x^{-1}}dx$ $\frac{df}{dn}$ should be sort of logarithmic and probably not interesting.? What is it, though? What have I tried? To see how far it is from something interesting, and maybe more..

Integral $\int \sqrt{x+\sqrt{x^2+2}}dx$

This question already has an answer here: Integrate $\int\sqrt{x+\sqrt{x^{2}+2}} dx$ . 3 answers

Show that $1/\sqrt{1} + 1/\sqrt{2} + … + 1/\sqrt{n} \leq 2\sqrt{n}-1$

This question already has an answer here: How to prove the inequality $2\sqrt{n + 1} − 2 \le 1 +\frac 1 {\sqrt 2}+\frac 1 {\sqrt 3}+ \dots +\frac 1 {\sqrt n} \le 2\sqrt n − 1$? 3 answers

Show that $(\sqrt{2}-1)^n$ is irrational

Show that for all $n\in \mathbb{N}$ the number $(\sqrt{2}-1)^n$ is irrational. I do not get the idea of the proof at all, any help appreaciated. edit: I am also thinking whether it will be possible to show $(\sqrt{2}-1)^n=\sqrt{m+1}-\sqrt{m}$