tìm giới hạn : A= $\lim_{n \to 1} \frac{\sqrt[]{2x-1}- x}{x^{2}-1 }$ B= $\lim_{n \to 2} \frac{\sqrt[3]{3x+2}- x}{\sqrt[]{3x-2}- 2 }$ C=$\lim_{n \to

a) $A =\lim\limits_{x\to 1}\dfrac{\sqrt{2x-1} -x}{x^2 -1}$

$\to A =\lim\limits_{x\to 1}\dfrac{(\sqrt{2x-1} -x)(\sqrt{2x-1} +x)}{(x-1)(x+1)(\sqrt{2x-1} +x)}$

$\to A =\lim\limits_{x\to 1}\dfrac{2x -1 – x^2}{(x-1)(x+1)(\sqrt{2x-1} +x)}$

$\to A =\lim\limits_{x\to 1}\dfrac{-(x-1)^2}{(x-1)(x+1)(\sqrt{2x-1} +x)}$

$\to A =\lim\limits_{x\to 1}\dfrac{1-x}{(x+1)(\sqrt{2x-1} +x)}$

$\to A =\dfrac{1-1}{(1+1)(\sqrt{2 -1} +1)}$

$\to A = 0$

b) $B =\lim\limits_{x\to 2}\dfrac{\sqrt[3]{3x+2} – x}{\sqrt{3x -2} -2}$

$\to B =\lim\limits_{x\to 2}\dfrac{(\sqrt[3]{3x+2} – x)(\sqrt[3]{(3x+2)^2} + x\sqrt[3]{3x+2} + x^2)(\sqrt{3x -2} +2)}{(\sqrt{3x -2} -2)(\sqrt{3x -2} +2)(\sqrt[3]{(3x+2)^2} + x\sqrt[3]{3x+2} + x^2)}$

$\to B =\lim\limits_{x\to 2}\dfrac{(3x +2 – x^3)(\sqrt{3x -2} +2)}{(3x -6)(\sqrt[3]{(3x+2)^2} + x\sqrt[3]{3x+2} + x^2)}$

$\to B =\lim\limits_{x\to 2}\dfrac{-(x-2)(x+1)^2(\sqrt{3x -2} +2)}{3(x -2)(\sqrt[3]{(3x+2)^2} + x\sqrt[3]{3x+2} + x^2)}$

$\to B =\lim\limits_{x\to 2}\dfrac{-(x+1)^2(\sqrt{3x -2} +2)}{3(\sqrt[3]{(3x+2)^2} + x\sqrt[3]{3x+2} + x^2)}$

$\to B =\dfrac{-(2+1)^2(\sqrt{3.2 -2} +2)}{3(\sqrt[3]{(3.2+2)^2} + 2.\sqrt[3]{3.2+2} + 2^2)}$

$\to B =\dfrac{-9.4}{3.(4 + 4 + 4)}$

$\to B = -1$

c) $C =\lim\limits_{x\to 1}\dfrac{\sqrt[3]{2x-1} -1}{x-1}$

$\to C =\lim\limits_{x\to 1}\dfrac{(\sqrt[3]{2x-1} -1)(\sqrt[3]{(2x-1)^2} + \sqrt[3]{2x -1} + 1)}{(x-1)(\sqrt[3]{(2x-1)^2} + \sqrt[3]{2x -1} + 1)}$

$\to C =\lim\limits_{x\to 1}\dfrac{2x -2}{(x-1)(\sqrt[3]{(2x-1)^2} + \sqrt[3]{2x -1} + 1)}$

$\to C =\lim\limits_{x\to 1}\dfrac{2}{\sqrt[3]{(2x-1)^2} + \sqrt[3]{2x -1} + 1}$

$\to C =\dfrac{2}{\sqrt[3]{(2-1)^2} + \sqrt[3]{2 -1} + 1}$

$\to C =\dfrac{2}{1+1+1}$

$\to C =\dfrac23$

d) $D =\lim\limits_{x\to 2}\dfrac{\sqrt[3]{3x+2} -2}{x^2 – 4}$

$\to D =\lim\limits_{x\to 2}\dfrac{(\sqrt[3]{3x+2} -2)(\sqrt[3]{(3x+2)^2} + 2\sqrt[3]{3x+2} + 4)}{(x-2)(x+2)(\sqrt[3]{(3x+2)^2} + 2\sqrt[3]{3x+2} + 4)}$

$\to D =\lim\limits_{x\to 2}\dfrac{3x -6 }{(x-2)(x+2)(\sqrt[3]{(3x+2)^2} + 2\sqrt[3]{3x+2} + 4)}$

$\to D =\lim\limits_{x\to 2}\dfrac{3}{(x+2)(\sqrt[3]{(3x+2)^2} + 2\sqrt[3]{3x+2} + 4)}$

$\to D =\dfrac{3}{(2+2)(\sqrt[3]{(3.2+2)^2} + 2\sqrt[3]{3.2+2} + 4)}$

$\to D=\dfrac{3}{4(4 + 4 +4)}$

$\to D =\dfrac{1}{16}$

e) $E =\lim\limits_{x\to 1}\dfrac{\sqrt[3]{7x+1} -\sqrt{5x-1}}{x-1}$

$\to E =\lim\limits_{x\to 1}\dfrac{\sqrt[3]{7x+1}-2+2 -\sqrt{5x-1}}{x-1}$

$\to E=\lim\limits_{x\to 1}\dfrac{\sqrt[3]{7x +1} -2}{x-1} + \lim\limits_{x\to 1}\dfrac{2 -\sqrt{5x-1}}{x-1}$

$\to E =\lim\limits_{x\to 1}\dfrac{(\sqrt[3]{7x +1} -2)(\sqrt[3]{(7x+1)^2} + 2\sqrt[3]{7x+1} + 4)}{(x-1)(\sqrt[3]{(7x+1)^2} + 2\sqrt[3]{7x+1} + 4)} + \lim\limits_{x\to 1}\dfrac{(2 -\sqrt{5x-1})(2 +\sqrt{5x-1})}{(x-1)(2 +\sqrt{5x-1})}$

$\to E =\lim\limits_{x\to 1}\dfrac{7x-7}{(x-1)(\sqrt[3]{(7x+1)^2} + 2\sqrt[3]{7x+1} + 4)} + \lim\limits_{x\to 1}\dfrac{5 – 5x}{(x-1)(2 +\sqrt{5x-1})}$

$\to E =\lim\limits_{x\to 1}\dfrac{7}{\sqrt[3]{(7x+1)^2} + 2\sqrt[3]{7x+1} + 4} + \lim\limits_{x\to 1}\dfrac{-1}{2 +\sqrt{5x-1}}$

$\to E =\dfrac{7}{\sqrt[3]{(7+1)^2} + 2\sqrt[3]{7+1} + 4} + \dfrac{-1}{2 +\sqrt{5-1}}$

$\to E =\dfrac{7}{4+4+4} -\dfrac{1}{2+2}$

$\to E = \dfrac13$

f) $F=\lim\limits_{x\to 7}\dfrac{\sqrt{x+2} -\sqrt[3]{x+20}}{\sqrt{x+9} -4}$

$\to F=\lim\limits_{x\to 7}\dfrac{\sqrt{x+2} -3+3-\sqrt[3]{x+20}}{\sqrt{x+9} -4}$

$\to F = \lim\limits_{x\to 7}\dfrac{\sqrt{x+2} -3}{\sqrt{x+9} -4} + \lim\limits_{x\to 7}\dfrac{3-\sqrt[3]{x+20}}{\sqrt{x+9} -4}$

$\to F =\lim\limits_{x\to 7}\dfrac{(\sqrt{x+2} -3)(\sqrt{x+2} +3)(\sqrt{x+9} +4)}{(\sqrt{x+9}-4)(\sqrt{x+9}+4)(\sqrt{x+2} +3)} + \lim\limits_{x\to 7}\dfrac{(3-\sqrt[3]{x+20})(9 + 3\sqrt[3]{x+20} + \sqrt[3]{(x+20)^2})(\sqrt{x+9}+4)}{(9 + 3\sqrt[3]{x+20} + \sqrt[3]{(x+20)^2})(\sqrt{x+9}+4)(\sqrt{x+9} -4)}$

$\to F =\lim\limits_{x\to 7}\dfrac{(x-7)(\sqrt{x+9}+4)}{(x-7)(\sqrt{x+2}+3)} +\lim\limits_{x\to 7}\dfrac{(7-x)(\sqrt{x+2}+3)}{(x-7)(9 + 3\sqrt[3]{x+20} + \sqrt[3]{(x+20)^2})}$

$\to F = \lim\limits_{x\to 7}\dfrac{\sqrt{x+9}+4}{\sqrt{x+2}+3} -\lim\limits_{x\to 7}\dfrac{\sqrt{x+2}+3}{9 + 3\sqrt[3]{x+20} + \sqrt[3]{(x+20)^2}}$

$\to F = \dfrac{\sqrt{7+9}+4}{\sqrt{7+2}+3} -\dfrac{\sqrt{7+2}+3}{9 + 3\sqrt[3]{7+20} + \sqrt[3]{(7+20)^2}}$

$\to F =\dfrac{4+4}{3+3} -\dfrac{3+3}{9+9 + 9}$

$\to F = \dfrac{10}{9}$

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