Fungrim entry: 14cbeb

\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 \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
`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("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))))

Topics using this entry

Square roots