Fungrim entry: 02751f

$\left|\sqrt{x + a} - \sqrt{x}\right| \le \sqrt{x} \left(1 - \sqrt{1 - \frac{\left|a\right|}{x}}\right)$
Assumptions:$x \in \left(0, \infty\right) \,\mathbin{\operatorname{and}}\, a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left|a\right| \le x$
TeX:
\left|\sqrt{x + a} - \sqrt{x}\right| \le \sqrt{x} \left(1 - \sqrt{1 - \frac{\left|a\right|}{x}}\right)

x \in \left(0, \infty\right) \,\mathbin{\operatorname{and}}\, a \in \mathbb{R} \,\mathbin{\operatorname{and}}\, \left|a\right| \le x
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
Sqrt$\sqrt{z}$ Principal square root
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("02751f"),
Formula(LessEqual(Abs(Sub(Sqrt(Add(x, a)), Sqrt(x))), Mul(Sqrt(x), Sub(1, Sqrt(Sub(1, Div(Abs(a), x))))))),
Variables(x, a),
Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(a, RR), LessEqual(Abs(a), x))))

Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2019-08-21 11:44:15.926409 UTC