Limx- 1-x2-2-x-1x-12 is equal to

Lim_x→ 1√(x^2) 2·√(x)+1(x 1)^2 = lim_x→ 1(√(x) 1)^2(x 1)^2 = lim_x→ 1(√(x) 1)^2((√(x))^3 1^3)^2 = lim_x→ 1((√(x) 1)(√(x) 1)(x^2/3+x^1/3+1))^2 =13^2=19 ...

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Standard XII

Mathematics

Rationalization Method to Remove Indeterminate Form

limx→ 1√x2-2·...

Question

$lim x \to 1 \frac{\sqrt[3]{x^{2}} - 2 \cdot \sqrt[3]{x} + 1}{(x - 1)^{2}}$ is equal to

\frac{1}{9}

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\frac{1}{3}

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\frac{1}{6}

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limit doesn't exist

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Solution

The correct option is A $\frac{1}{9}$
$lim x \to 1 \frac{\sqrt[3]{x^{2}} - 2 \cdot \sqrt[3]{x} + 1}{(x - 1)^{2}}$

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

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

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

$= \frac{1}{3^{2}} = \frac{1}{9}$

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Similar questions

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

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lim x \to 1 \frac{\sqrt[3]{x^{2}} - 2. \sqrt[3]{x} + 1}{(x - 1)^{2}}

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Q. If

f (x) = \frac{2}{x - 3}, g (x) = \frac{x - 3}{x + 4}

and

h (x) = \frac{- 2 (2 x + 1)}{x^{2} + x - 12}

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lim x \to 3 [f (x) + g (x) + h (x)]

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Rationalization Method to Remove Indeterminate Form

Standard XII Mathematics

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limx→13√x2−2⋅3√x+1(x−1)2 is equal to

The correct option is A 19limx→13√x2−2⋅3√x+1(x−1)2 =limx→1(3√x−1)2(x−1)2 =limx→1(3√x−1)2((3√x)3−13)2 =limx→1((3√x−1)(3√x−1)(x2/3+x1/3+1))2 =132=19

$lim x \to 1 \frac{\sqrt[3]{x^{2}} - 2 \cdot \sqrt[3]{x} + 1}{(x - 1)^{2}}$ is equal to