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

Square roots