query: Fix $a > 0$ and let $x_1 > \sqrt a$
positive_passages.docid: 31174
positive_passages.text: Find the limit if it exists of $S_{n+1} = \frac{1}{2}(S_n +\frac{A}{S_n})$
negative_passages.docid: 128091
negative_passages.text: Find the range of possible values of $\sqrt{a^2+a+1}-\sqrt{a^2-a+1}$
negative_passages.docid: 151593
negative_passages.text: Prove that the limit of $\sqrt{n+1}-\sqrt{n}$ is zero
negative_passages.docid: 91877
negative_passages.text: Let $|x|\leq1, |y|\leq 1$. Show that$0 \leq x^{2}+y^{2} -2x^{2} y^{2}+2xy \sqrt{1-x^{2}} \sqrt{1-y^{2}} \leq 1$
negative_passages.docid: 82775
negative_passages.text: if $a>1$, Prove that $\lim a^{1\over n}=1$
negative_passages.docid: 110661
negative_passages.text: When does $\sqrt{a b} = \sqrt{a} \sqrt{b}$?
negative_passages.docid: 90199
negative_passages.text: Prove that there exists a constant $c>0$ such that the following happens
negative_passages.docid: 156034
negative_passages.text: sufficient condition for a polynomial to have roots in $[0,1]$
negative_passages.docid: 111432
negative_passages.text: Square root of a number squared is equal to the absolute value of that number
negative_passages.docid: 95293
negative_passages.text: Proof that $|\sqrt{x}-\sqrt{y}| \leq \sqrt{|x-y|},\quad x,y \geq 0$
negative_passages.docid: 37394
negative_passages.text: How to find $\lim\limits_{n\to\infty} n·(\sqrt[n]{a}-1)$?
negative_passages.docid: 17343
negative_passages.text: Function on $[a,b]$ that satisfies a Hölder condition of order $\alpha > 1 $ is constant
negative_passages.docid: 132610
negative_passages.text: Expand $\sqrt{1-x}$ up to and including the term $x^2$ (How should i proceed further)
negative_passages.docid: 77026
negative_passages.text: Proof with inequalities
negative_passages.docid: 79925
negative_passages.text: Infinitely many values of $x$ that satisfy the equations
negative_passages.docid: 89325
negative_passages.text: $a_{m+n}\leq a_ma_n$ implies the convergence of $\sqrt[n]a_n$
negative_passages.docid: 88640
negative_passages.text: Is $\sqrt x$ continuous at $0$? Because it is not defined to the left of $0$