limx→a (a+2x)1/3−(3x)1/3(3a+x)1/3−(4x)1/3,(a≠0) is equal to

A
$\frac{2}{9} {(\frac{2}{3})}^{1 / 3}$
B
${(\frac{2}{3})}^{1 / 3}$
C
${(\frac{2}{9})}^{4 / 3}$
D
$\frac{2}{3} {(\frac{2}{9})}^{1 / 3}$

Solution:

Let $L = lim_{x \to a} \frac{(a + 2 x)^{1 / 3} - (3 x)^{1 / 3}}{(3 a + x)^{1 / 3} - (4 x)^{1 / 3}} .$ Applying $L'$

Hospital's rule, we obtain

$\begin{array}{l} L = lim_{x \to a} \frac{\frac{2}{3} (a + 2 x)^{- 2 / 3} - (3 x)^{- 2 / 3}}{\frac{1}{3} (3 a + x)^{- 2 / 3} - \frac{4}{3} (4 x)^{- 2 / 3}} 1 \\ \Rightarrow L = \frac{- \frac{1}{3} (3 a)^{- 2 / 3}}{- (4 a)^{- 2 / 3}} \\ \Rightarrow L = \frac{1}{3} \frac{(4 a)^{2 / 3}}{(3 a)^{2 / 3}} = \frac{2^{4 / 3}}{3^{5 / 3}} = \frac{2}{3} {(\frac{2}{9})}^{1 / 3} \end{array}$

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

Solution:

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