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Fungrim entry: 14cbeb

x+axx2(ax+a2x2)\left|\sqrt{x + a} - \sqrt{x}\right| \le \frac{\sqrt{x}}{2} \left(\frac{\left|a\right|}{x} + \frac{{\left|a\right|}^{2}}{{x}^{2}}\right)
Assumptions:x(0,)andaRandaxx \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 \frac{\sqrt{x}}{2} \left(\frac{\left|a\right|}{x} + \frac{{\left|a\right|}^{2}}{{x}^{2}}\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
Absz\left|z\right| Absolute value
Sqrtz\sqrt{z} Principal square root
Powab{a}^{b} Power
OpenInterval(a,b)\left(a, b\right) Open interval
Infinity\infty Positive infinity
RRR\mathbb{R} Real numbers
Source code for this entry:
Entry(ID("14cbeb"),
    Formula(LessEqual(Abs(Sub(Sqrt(Add(x, a)), Sqrt(x))), Mul(Div(Sqrt(x), 2), Add(Div(Abs(a), x), Div(Pow(Abs(a), 2), Pow(x, 2)))))),
    Variables(x, a),
    Assumptions(And(Element(x, OpenInterval(0, Infinity)), Element(a, RR), LessEqual(Abs(a), x))))

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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