Limx- aa-2x 1-3-3x 1-33a-x 1-3-4x 1-3 a- 0 is equal to-

Lim_x→ a(a+2x )^1/3 (3x )^1/3(3a+x )^1/3 (4x )^1/3 (00 form ) Applying L'Hospital's Rule =lim_x→ a23(a+2x )^ 23 3^13·13x^ 2313(3a+x )^ 23 4^13·13x^ 23 = ...

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Byju's Answer

Standard XII

Mathematics

Continuity of a Function

limx→ aa+2x 1...

Question

$lim x \to a \frac{{(a + 2 x)}^{\frac{1}{3}} - {(3 x)}^{\frac{1}{3}}}{{(3 a + x)}^{\frac{1}{3}} - {(4 x)}^{\frac{1}{3}}} (a \neq 0)$ is equal to:

{(\frac{2}{9})}^{\frac{4}{3}}

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{(\frac{2}{3})}^{\frac{4}{3}}

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(\frac{2}{3}) {(\frac{2}{9})}^{\frac{1}{3}}

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(\frac{2}{9}) {(\frac{2}{3})}^{\frac{1}{3}}

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Solution

The correct option is C $(\frac{2}{3}) {(\frac{2}{9})}^{\frac{1}{3}}$
$lim x \to a \frac{{(a + 2 x)}^{\frac{1}{3}} - {(3 x)}^{\frac{1}{3}}}{{(3 a + x)}^{\frac{1}{3}} - {(4 x)}^{\frac{1}{3}}} (\frac{0}{0} form)$
Applying L'Hospital's Rule
$= lim x \to a \frac{\frac{2}{3} {(a + 2 x)}^{\frac{- 2}{3}} - 3^{\frac{1}{3}} \cdot \frac{1}{3} x^{- \frac{2}{3}}}{\frac{1}{3} {(3 a + x)}^{- \frac{2}{3}} - 4^{\frac{1}{3}} \cdot \frac{1}{3} x^{- \frac{2}{3}}}$
$= \frac{\frac{2}{3} {(3 a)}^{- \frac{2}{3}} - 3^{- \frac{2}{3}} \cdot ⎛ ⎜ ⎝ a^{- \frac{2}{3}} ⎞ ⎟ ⎠}{\frac{1}{3} {(4 a)}^{- \frac{2}{3}} - \frac{1}{3} \cdot 4^{\frac{1}{3}} ⎛ ⎜ ⎝ a^{- \frac{2}{3}} ⎞ ⎟ ⎠}$
$= \frac{2}{3} \cdot {(\frac{2}{9})}^{\frac{1}{3}}$

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limx→a(a+2x)13−(3x)13(3a+x)13−(4x)13 (a≠0) is equal to:

The correct option is C (23)(29)13 limx→a(a+2x)13−(3x)13(3a+x)13−(4x)13 (00 form) Applying L'Hospital's Rule =limx→a23(a+2x)−23−313⋅13x−2313(3a+x)−23−413⋅13x−23 =23(3a)−23−3−23⋅⎛⎜⎝a−23⎞⎟⎠13(4a)−23−13⋅413⎛⎜⎝a−23⎞⎟⎠ =23⋅(29)13

$lim x \to a \frac{{(a + 2 x)}^{\frac{1}{3}} - {(3 x)}^{\frac{1}{3}}}{{(3 a + x)}^{\frac{1}{3}} - {(4 x)}^{\frac{1}{3}}} (a \neq 0)$ is equal to: