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دالة الجذر من الرتبة n

تمرين1

اجعل مقامات الاعداد التالية اعدادا صحيحية طبيعية
$\frac{1}{\sqrt[3]{3} - \sqrt[3]{2}}; \frac{1}{\sqrt[4]{5} - \sqrt[4]{2}}; \frac{1}{1 + \sqrt[3]{3} + \sqrt[3]{9}}$
قارن الاعداد التالية
$\sqrt[3]{4}; \sqrt[]{2}; \sqrt[6]{7}; \sqrt[12]{3}$
حل في IR المعادلات التالية:
$x^{3} = - 8; {(x + 1)}^{3} = 2; {(x - 1)}^{3} = - 4$
حل في IR المتراجحات التالية:
$\sqrt[3]{x - 2} 〉 1; \sqrt[3]{1 - 2 x} 〉 2; \sqrt[4]{2 - 3 x} \leq 2$

تمرين 2

احسب النهايات التالية $\begin{array}{l} \lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{x - 1}; \lim_{x \to 8} \frac{x - 8}{\sqrt[3]{x} - 2}; \lim_{x \to - 6} \frac{\sqrt[3]{2 - x} - 2}{x^{2} + 7 x + 6}; \lim_{x \to + \infty} \sqrt[3]{x + 4} - \sqrt[3]{x}; \\ \lim_{x \to + \infty} \sqrt[]{x^{3} + x + 1} - \sqrt[3]{2 x^{3} + 2}; \lim_{x \to + \infty} \sqrt[3]{x^{3} + x + 1} - \sqrt[3]{x^{3} + 1}; \\ \lim_{x \to + \infty} \frac{\sqrt[4]{x + 1} - \sqrt[3]{x + 1}}{\sqrt{x + 1} + \sqrt[6]{x + 1}} (t = \sqrt[12]{x + 1}) \end{array}$

$= \frac{1}{1^{2} + (1 \times \sqrt[3]{3}) + {(\sqrt[3]{3})}^{2}}$	$\frac{1}{1 + \sqrt[3]{3} + \sqrt[3]{9}}$
$= \frac{(\sqrt[3]{3} - 1)}{(\sqrt[3]{3} - 1) [{(\sqrt[3]{3})}^{2} + (1 \times \sqrt[3]{3}) + 1^{2}]}$
$= \frac{\sqrt[3]{3} - 1}{{(\sqrt[3]{3})}^{3} - 1^{3}}$
$= \frac{\sqrt[3]{3} - 1}{3 - 1}$
$= \frac{\sqrt[3]{3} - 1}{2}$

$= \frac{(\sqrt[4]{5} + \sqrt[4]{2})}{(\sqrt[4]{5} - \sqrt[4]{2}) (\sqrt[4]{5} + \sqrt[4]{2})}$	$\frac{1}{\sqrt[4]{5} - \sqrt[4]{2}}$
$= \frac{(\sqrt[4]{5} + \sqrt[4]{2})}{\sqrt{5} - \sqrt{2}}$
$= \frac{(\sqrt[4]{5} + \sqrt[4]{2}) (\sqrt{5} + \sqrt{2})}{(\sqrt{5} - \sqrt{2}) (\sqrt{5} + \sqrt{2})}$
$= \frac{(\sqrt[4]{5} + \sqrt[4]{2}) (\sqrt{5} + \sqrt{2})}{5 - 2}$
$= \frac{(\sqrt[4]{5} + \sqrt[4]{2}) (\sqrt{5} + \sqrt{2})}{3}$

$= \frac{{(\sqrt[3]{3})}^{2} + (\sqrt[3]{3} \times \sqrt[3]{2}) + {(\sqrt[3]{2})}^{2}}{(\sqrt[3]{3} - \sqrt[3]{2}) [{(\sqrt[3]{3})}^{2} + (\sqrt[3]{3} \times \sqrt[3]{2}) + {(\sqrt[3]{2})}^{2}]}$	$\frac{1}{\sqrt[3]{3} - \sqrt[3]{2}}$
$= \frac{{(\sqrt[3]{3})}^{2} + (\sqrt[3]{3} \times \sqrt[3]{2}) + {(\sqrt[3]{2})}^{2}}{{(\sqrt[3]{3})}^{3} - {(\sqrt[3]{2})}^{3}}$
$= \frac{{(\sqrt[3]{3})}^{2} + (\sqrt[3]{3} \times \sqrt[3]{2}) + {(\sqrt[3]{2})}^{2}}{3 - 2}$
$= {(\sqrt[3]{3})}^{2} + (\sqrt[3]{3} \times \sqrt[3]{2}) + {(\sqrt[3]{2})}^{2}$

${(x - 1)}^{3} = 4$	${(x - 1)}^{3} = {(\sqrt[3]{4})}^{3} \Leftrightarrow$
	$x - 1 = \sqrt[3]{4} \Leftrightarrow$
	$x = \sqrt[3]{4} + 1 \Leftrightarrow$

$= {x \in ℝ / 2 - 3 x \geq 0}$	$D$
$= {x \in ℝ / x \leq \frac{2}{3}}$
$=] - \infty; \frac{2}{3}]$

$= \lim_{x \to 1} \frac{(\sqrt[3]{x} - 1) ({(\sqrt[3]{x})}^{2} + \sqrt[3]{x} + 1^{2})}{(x - 1) ({(\sqrt[3]{x})}^{2} + \sqrt[3]{x} + 1^{2})}$	$\lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{x - 1}$
$= \lim_{x \to 1} \frac{(x - 1)}{(x - 1) ({(\sqrt[3]{x})}^{2} + \sqrt[3]{x} + 1^{2})}$
$= \lim_{x \to 1} \frac{1}{({(\sqrt[3]{x})}^{2} + \sqrt[3]{x} + 1^{2})}$
$= \frac{1}{3}$

$= \lim_{x \to 8} \frac{{(\sqrt[3]{x})}^{3} - 2^{3}}{\sqrt[3]{x} - 2}$	$\lim_{x \to 8} \frac{x - 8}{\sqrt[3]{x} - 2}$
$= \lim_{x \to 8} \frac{(\sqrt[3]{x} - 2) ({(\sqrt[3]{x})}^{2} + 2 \sqrt[3]{x} + 4)}{\sqrt[3]{x} - 2}$
$= \lim_{x \to 8} {(\sqrt[3]{x})}^{2} + 2 \sqrt[3]{x} + 4$
$= 12$

$= \lim_{x \to - 6} \frac{(\sqrt[3]{2 - x} - 2) ({(\sqrt[3]{2 - x})}^{2} + 2 \sqrt[3]{2 - x} + 4)}{(x^{2} + 7 x + 6) ({(\sqrt[3]{2 - x})}^{2} + 2 \sqrt[3]{2 - x} + 4)}$	$\lim_{x \to - 6} \frac{\sqrt[3]{2 - x} - 2}{x^{2} + 7 x + 6}$
$= \lim_{x \to - 6} \frac{{(\sqrt[3]{2 - x})}^{3} - 2^{3}}{(x + 6) (x + 1) ({(\sqrt[3]{2 - x})}^{2} + 2 \sqrt[3]{2 - x} + 4)}$
$= \lim_{x \to - 6} \frac{(2 - x) - 8}{(x + 6) (x + 1) ({(\sqrt[3]{2 - x})}^{2} + 2 \sqrt[3]{2 - x} + 4)}$
$= \lim_{x \to - 6} \frac{- (x + 6)}{(x + 6) (x + 1) ({(\sqrt[3]{2 - x})}^{2} + 2 \sqrt[3]{2 - x} + 4)}$
$= \lim_{x \to - 6} \frac{- 1}{(x + 1) ({(\sqrt[3]{2 - x})}^{2} + 2 \sqrt[3]{2 - x} + 4)}$
$= \frac{1}{60}$

$= \lim_{x \to + \infty} \frac{(\sqrt[3]{x + 4} - \sqrt[3]{x}) ({(\sqrt[3]{x + 4})}^{2} + \sqrt[3]{x + 4} \sqrt[3]{x} + {(\sqrt[3]{x})}^{2})}{({(\sqrt[3]{x + 4})}^{2} + \sqrt[3]{x + 4} \sqrt[3]{x} + {(\sqrt[3]{x})}^{2})}$	$\lim_{x \to + \infty} \sqrt[3]{x + 4} - \sqrt[3]{x}$
$= \lim_{x \to + \infty} \frac{{(\sqrt[3]{x + 4})}^{3} - {(\sqrt[3]{x})}^{3}}{({(\sqrt[3]{x + 4})}^{2} + \sqrt[3]{x^{2} + 4 x} + {(\sqrt[3]{x})}^{2})}$
$= \lim_{x \to + \infty} \frac{(x + 4) - (x)}{({(\sqrt[3]{x + 4})}^{2} + \sqrt[3]{x^{2} + 4 x} + {(\sqrt[3]{x})}^{2})}$
$= \lim_{x \to + \infty} \frac{4}{({(\sqrt[3]{x + 4})}^{2} + \sqrt[3]{x^{2} + 4 x} + {(\sqrt[3]{x})}^{2})}$
$= 0$

$= \lim_{x \to + \infty} x (\sqrt{x + \frac{1}{x} + \frac{1}{x^{2}}} - \sqrt[3]{2 + \frac{2}{x^{3}}})$	$\lim_{x \to + \infty} \sqrt{x^{3} + x + 1} - \sqrt[3]{2 x^{3} + 2}$
$= + \infty$

$= \lim_{x \to + \infty} \frac{(x^{3} + x + 1) - (x^{3} + 1)}{{(\sqrt[3]{x^{3} + x + 1})}^{2} + \sqrt[3]{x^{3} + x + 1} \sqrt[3]{x^{3} + 1} + {(\sqrt[3]{x^{3} + 1})}^{2}}$	$\lim_{x \to + \infty} \sqrt[3]{x^{3} + x + 1} - \sqrt[3]{x^{3} + 1}$
$= \lim_{x \to + \infty} \frac{x}{x^{2} [{(\sqrt[3]{1 + \frac{1}{x^{2}} + \frac{1}{x^{3}}})}^{2} + \sqrt[3]{1 + \frac{1}{x^{2}} + \frac{1}{x^{3}}} \sqrt[3]{1 + \frac{1}{x^{3}}} + {(\sqrt[3]{1 + \frac{1}{x^{3}}})}^{2}]}$
$= \lim_{x \to + \infty} \frac{1}{x [{(\sqrt[3]{1 + \frac{1}{x^{2}} + \frac{1}{x^{3}}})}^{2} + \sqrt[3]{1 + \frac{1}{x^{2}} + \frac{1}{x^{3}}} \sqrt[3]{1 + \frac{1}{x^{3}}} + {(\sqrt[3]{1 + \frac{1}{x^{3}}})}^{2}]}$
$= 0$

$= \lim_{x \to + \infty} \frac{\sqrt[12]{{(x + 1)}^{3}} - \sqrt[12]{{(x + 1)}^{4}}}{\sqrt[12]{{(x + 1)}^{6}} + \sqrt[12]{{(x + 1)}^{2}}}$	$\lim_{x \to + \infty} \frac{\sqrt[4]{x + 1} - \sqrt[3]{x + 1}}{\sqrt{x + 1} + \sqrt[6]{x + 1}}$
$= \lim_{x \to + \infty} \frac{\sqrt[12]{(x + 1)} - \sqrt[12]{{(x + 1)}^{2}}}{\sqrt[12]{{(x + 1)}^{4}} + 1}$
$= \lim_{t \to + \infty} \frac{t - t^{2}}{t^{4} + 1}$
$= \lim_{t \to + \infty} \frac{\frac{1}{t} - 1}{t^{2} + \frac{1}{t^{2}}}$
$= 0$

$= \lim_{x \to 16} \frac{(\sqrt[4]{x} - 2) (\sqrt[4]{x} + 2)}{(\sqrt{x} - 4) (\sqrt[4]{x} + 2)}$	$\lim_{x \to 16} \frac{\sqrt[4]{x} - 2}{\sqrt{x} - 4}$
$= \lim_{x \to 16} \frac{\sqrt{x} - 4}{(\sqrt{x} - 4) (\sqrt[4]{x} + 2)}$
$= \lim_{x \to 16} \frac{1}{(\sqrt[4]{x} + 2)}$
$= \frac{1}{4}$

$= \lim_{x \to + \infty} \frac{\sqrt{x} (1 + \frac{1}{\sqrt[6]{x}} + \frac{1}{\sqrt[4]{x}})}{\sqrt{x} (\sqrt{2 + \frac{1}{x}})}$	$\lim_{x \to + \infty} \frac{\sqrt{x} + \sqrt[3]{x} + \sqrt[4]{x}}{\sqrt{2 x + 1}}$
$= \lim_{x \to + \infty} \frac{1 + \frac{1}{\sqrt[6]{x}} + \frac{1}{\sqrt[4]{x}}}{\sqrt{2 + \frac{1}{x}}}$
$= \frac{\sqrt{2}}{2}$

$= \lim_{x \to + \infty} \frac{x}{x - 1}$	$\lim_{x \to + \infty} \frac{x}{\sqrt[3]{x^{3}} - 1}$
$= \lim_{x \to + \infty} \frac{x}{x}$
$= 1$

$= \lim_{x \to + \infty} \sqrt[3]{x - 1} ((\sqrt[3]{x^{2} + x + 1}) - 1)$	$\lim_{x \to + \infty} \sqrt[3]{x^{3} - 1} - \sqrt[3]{x - 1}$
$= + \infty$

$= \lim_{x \to 1} \frac{(x - 1) ({(\sqrt[3]{x + 7})}^{2} + 2 \sqrt[3]{x + 7} + 4)}{[(x + 7) - 8] [{(\sqrt[3]{x})}^{2} + \sqrt[3]{x} + 1]}$	$\lim_{x \to 1} \frac{\sqrt[3]{x} - 1}{\sqrt[3]{x + 7} - 2}$
$= \lim_{x \to 1} \frac{(x - 1) ({(\sqrt[3]{x + 7})}^{2} + 2 \sqrt[3]{x + 7} + 4)}{[x - 1] [{(\sqrt[3]{x})}^{2} + \sqrt[3]{x} + 1]}$
$= \lim_{x \to 1} \frac{{(\sqrt[3]{x + 7})}^{2} + 2 \sqrt[3]{x + 7} + 4}{{(\sqrt[3]{x})}^{2} + \sqrt[3]{x} + 1}$
$= \frac{12}{3}$
$= 4$

$= \lim_{x \to + \infty} x (\sqrt[3]{\frac{1}{x} + \frac{1}{x^{3}}} - \sqrt[3]{1 - \frac{1}{x^{3}}})$	$\lim_{x \to + \infty} \sqrt[3]{x^{2} + 1} - \sqrt[3]{x^{3} - 1}$
$= - \infty$

$= \lim_{x \to 0} \frac{(3 x - x^{2})}{(x + x^{2}) ({(\sqrt[3]{8 + 3 x - x^{2}})}^{2} + 2 \sqrt[3]{8 + 3 x - x^{2}} + 4)}$	$\lim_{x \to 0} \frac{\sqrt[3]{8 + 3 x - x^{2}} - 2}{x + x^{2}}$
$= \lim_{x \to 0} \frac{(3 - x)}{(1 + x) ({(\sqrt[3]{8 + 3 x - x^{2}})}^{2} + 2 \sqrt[3]{8 + 3 x - x^{2}} + 4)}$
$= \frac{1}{4}$

$= \lim_{x \to + \infty} \frac{((x^{3} + x + 1) - x^{3})}{({(\sqrt[3]{x^{3} + x + 1})}^{2} + x \sqrt[3]{x^{3} + x + 1} + x^{2})}$	$\lim_{x \to + \infty} \sqrt[3]{x^{3} + x + 1} - x$
$= \lim_{x \to + \infty} \frac{x + 1}{({(\sqrt[3]{x^{3} + x + 1})}^{2} + x \sqrt[3]{x^{3} + x + 1} + x^{2})}$
$= \lim_{x \to + \infty} \frac{1 + \frac{1}{x}}{(x {(\sqrt[3]{1 + \frac{1}{x^{2}} + \frac{1}{x^{3}}})}^{2} + \sqrt[3]{x^{3} + x + 1} + x)}$
$= 0$

$= \lim_{x \to 0} \frac{((1 + x) - (1 - x)) ({(\sqrt[3]{1 + x})}^{2} + \sqrt[3]{1 + x} \sqrt[3]{1 - x} + {(\sqrt[3]{1 - x})}^{2})}{((1 + x) - (1 - x)) (\sqrt{1 + x} + \sqrt{1 - x})}$	$\lim_{x \to 0} \frac{\sqrt{1 + x} - \sqrt{1 - x}}{\sqrt[3]{1 + x} - \sqrt[3]{1 - x}}$
$= \lim_{x \to 0} \frac{({(\sqrt[3]{1 + x})}^{2} + \sqrt[3]{1 + x} \sqrt[3]{1 - x} + {(\sqrt[3]{1 - x})}^{2})}{(\sqrt{1 + x} + \sqrt{1 - x})}$
$= \frac{3}{2}$

$= \lim_{x \to 0} \frac{(27 + x) - (27 - x)}{x (1 + 2 \sqrt[3]{x}) ({(\sqrt[3]{27 + x})}^{2} + \sqrt[3]{27 + x} \sqrt[3]{27 - x} + {(\sqrt[3]{27 - x})}^{2})}$	$\lim_{x \to 0} \frac{\sqrt[3]{27 + x} - \sqrt[3]{27 - x}}{x + 2 \sqrt[3]{x^{4}}}$
$= \lim_{x \to 0} \frac{2 x}{x (1 + 2 \sqrt[3]{x}) ({(\sqrt[3]{27 + x})}^{2} + \sqrt[3]{27 + x} \sqrt[3]{27 - x} + {(\sqrt[3]{27 - x})}^{2})}$
$= \lim_{x \to 0} \frac{2}{(1 + 2 \sqrt[3]{x}) ({(\sqrt[3]{27 + x})}^{2} + \sqrt[3]{27 + x} \sqrt[3]{27 - x} + {(\sqrt[3]{27 - x})}^{2})}$
$= \frac{2}{27}$

$= \lim_{x \to 1} \frac{(2 x + 6) - 8}{(x - 1) ({(\sqrt[3]{2 x + 6})}^{2} + 2 \sqrt[3]{2 x + 6} + 4)}$	$\lim_{x \to 1} \frac{\sqrt[3]{2 x + 6} - 2}{x - 1}$
$\lim_{x \to 1} \frac{2 (x - 1)}{(x - 1) ({(\sqrt[3]{2 x + 6})}^{2} + 2 \sqrt[3]{2 x + 6} + 4)}$
$= \lim_{x \to 1} \frac{2}{({(\sqrt[3]{2 x + 6})}^{2} + 2 \sqrt[3]{2 x + 6} + 4)}$
$= \frac{2}{12}$
$= \frac{1}{6}$

$= \lim_{x \to + \infty} \frac{(x^{3} + x + 1) - (x^{3} + 2)}{({(\sqrt[3]{x^{3} + x + 1})}^{2} + \sqrt[3]{x^{3} + x + 1} \sqrt[3]{x^{3} + 2} + {(\sqrt[3]{x^{3} + 2})}^{2})}$	$\lim_{x \to + \infty} \sqrt[3]{x^{3} + x + 1} - \sqrt[3]{x^{3} + 2}$
$= \lim_{x \to + \infty} \frac{x - 1}{({(\sqrt[3]{x^{3} + x + 1})}^{2} + \sqrt[3]{x^{3} + x + 1} \sqrt[3]{x^{3} + 2} + {(\sqrt[3]{x^{3} + 2})}^{2})}$
$= \lim_{x \to + \infty} \frac{1 - \frac{1}{x}}{(x {(\sqrt[3]{1 + \frac{1}{x^{2}} + \frac{1}{x^{3}}})}^{2} + \sqrt[3]{x^{3} + x + 1} \sqrt[3]{1 + \frac{2}{x^{3}}} + x {(\sqrt[3]{1 + \frac{2}{x^{3}}})}^{2})}$
$= 0$

دالة الجذر من الرتبة n

تمرين1

تمرين 2

تمرين3

تمرين4

دالة قوس الظل

تمرين5

تمرين6

تمرين7

تمرين8

تمرين9

تمرين10

تمرين11

تمرين12

الدالة العكسية

تمرين13

تمرين14

تمرين15

تمرين16

تمرين17

تمرين18

تمرين19

تمرين20

تمرين21

تمرين22

تمرين23

تمرين24

تمرين25

عناصر الإجابة

$= \frac{\tan (A r c \tan (x)) - \tan (A r c \tan (y))}{1 + \tan (A r c \tan (x)) \times \tan (A r c \tan (y))}$	$\tan (A r c \tan (x) - A r c \tan (y))$
$= \frac{x - y}{1 + x y}$

$= \frac{\tan (A r c \tan (\frac{1}{x})) - \tan (A r c \tan (\frac{1}{y}))}{1 + \tan (A r c \tan (\frac{1}{x})) \times \tan (A r c \tan (\frac{1}{y}))}$	$\tan (A r c \tan (\frac{1}{x}) - A r c \tan (\frac{1}{y}))$
$= \frac{\frac{1}{x} - \frac{1}{y}}{1 + \frac{1}{x} \times \frac{1}{y}}$
$= \frac{y - x}{x y + 1}$

$= \frac{2 \tan (A r c \tan (x))}{1 - {(\tan (A c \tan (x)))}^{2}}$	$\tan (2 A r c \tan (x))$
$= \frac{2 x}{1 - x^{2}}$

$= \lim_{X \to + \infty} A r c \tan X$	$\lim_{x \to 1^{-}} A r c \tan \sqrt{\frac{1 + x^{2}}{1 - x^{2}}}$
$= \frac{π}{2}$

$= \lim_{X \to 1} A r c \tan X$	$\lim_{x \to + \infty} A r c \tan \sqrt{\frac{x - 1}{x + 1}}$
$= A r c \tan (1)$
$= \frac{π}{4}$

$= \lim_{X \to 0} \frac{X}{\tan (X + \frac{π}{6}) - \frac{\sqrt{3}}{3}}$	$\lim_{x \to \frac{\sqrt{3}}{3}} \frac{A r c \tan x - \frac{π}{6}}{x - \frac{\sqrt{3}}{3}}$
$= \lim_{X \to 0} \frac{X}{\frac{\tan X + \frac{\sqrt{3}}{3}}{1 - \frac{\sqrt{3}}{3} \tan X} - \frac{\sqrt{3}}{3}}$
$= \lim_{X \to 0} \frac{3}{4} (\frac{X}{\tan X}) (1 - \frac{\sqrt{3}}{3} \tan X)$
$= \frac{3}{4}$

$= \lim_{X \to - 1} \arctan X$	$\lim_{x \to + \infty} \arctan (\frac{{(1 + x)}^{2}}{1 - x^{2}})$
$= \arctan (- 1)$
$= - \arctan 1$
$= - \frac{π}{4}$

$= \lim_{X \to 0} \frac{X}{\frac{\tan X}{3}}$	$\lim_{x \to 0} \frac{\arctan 3 x}{x}$
$= \lim_{X \to 0} 3 (\frac{X}{\tan X})$
$= 3$

$] - \frac{1}{4}; + \infty [\to] \frac{1}{2}; + \infty [$	$h^{- 1} :$
$x \mapsto h^{- 1} (x) = y$

$\Leftrightarrow x = h (y)$	$h^{- 1} (x) = y$
$\Leftrightarrow y^{2} - y - x = 0$