# Fungrim entry: 34136c

$\left|\frac{1}{\sqrt{x + a}} - \frac{1}{\sqrt{x}}\right| \le \frac{\left|a\right|}{2 {\left(x - \left|a\right|\right)}^{3 / 2}}$
Assumptions:$x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|a\right| < x$
TeX:
\left|\frac{1}{\sqrt{x + a}} - \frac{1}{\sqrt{x}}\right| \le \frac{\left|a\right|}{2 {\left(x - \left|a\right|\right)}^{3 / 2}}

x \in \left(0, \infty\right) \;\mathbin{\operatorname{and}}\; a \in \mathbb{R} \;\mathbin{\operatorname{and}}\; \left|a\right| < x
Definitions:
Fungrim symbol Notation Short description
Abs$\left|z\right|$ Absolute value
Sqrt$\sqrt{z}$ Principal square root
Pow${a}^{b}$ Power
OpenInterval$\left(a, b\right)$ Open interval
Infinity$\infty$ Positive infinity
RR$\mathbb{R}$ Real numbers
Source code for this entry:
Entry(ID("34136c"),
Formula(LessEqual(Abs(Sub(Div(1, Sqrt(Add(x, a))), Div(1, Sqrt(x)))), Div(Abs(a), Mul(2, Pow(Sub(x, Abs(a)), Div(3, 2)))))),
Variables(x, a),
Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(a, RR), Less(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.

2021-03-15 19:12:00.328586 UTC