lim X--> 0((x+2)^1/3 - (2)^1/3)/x

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$$\begin{gathered} \mathrm{We} \mathrm{have}: \\ \mathrm{L}=\underset{\mathrm{x}\to 0}{\lim } \frac{{\left(\mathrm{x}+2\right)}^{{\displaystyle \frac{1}{3}}}-2^{{\displaystyle \frac{1}{3}}}}{\mathrm{x}} \\ =\underset{\mathrm{x}\to 0}{\lim } \left[\frac{{\left(\mathrm{x}+2\right)}^{{\displaystyle \frac{1}{3}}}-2^{{\displaystyle \frac{1}{3}}}}{\mathrm{x}}\right]\left[\frac{{\left(\mathrm{x}+2\right)}^{{\displaystyle \frac{2}{3}}}+2^{{\displaystyle \frac{2}{3}}}+{\left(\mathrm{x}+2\right)}^{{\displaystyle \frac{1}{3}}}.2^{{\displaystyle \frac{1}{3}}}}{{\left(\mathrm{x}+2\right)}^{\frac{2}{3}}+2^{\frac{2}{3}}+{\left(\mathrm{x}+2\right)}^{\frac{1}{3}}.2^{\frac{1}{3}}}\right] \\ \mathrm{We} \mathrm{know} \mathrm{that}: \\ \left(\mathrm{a}-\mathrm{b}\right)\left({\mathrm{a}}^2+{\mathrm{b}}^2+\mathrm{ab}\right)={\mathrm{a}}^3-{\mathrm{b}}^3, \mathrm{hence} \mathrm{we} \mathrm{get}: \\ \mathrm{L}=\underset{\mathrm{x}\to 0}{\lim } \left[\frac{{\left(\mathrm{x}+2\right)}^{{\displaystyle \frac{3}{3}}}-2^{{\displaystyle \frac{3}{3}}}}{\mathrm{x}}\right]\left[\frac{1}{{\left(\mathrm{x}+2\right)}^{\frac{2}{3}}+2^{\frac{2}{3}}+{\left(\mathrm{x}+2\right)}^{\frac{1}{3}}.2^{\frac{1}{3}}}\right] \\ =\underset{\mathrm{x}\to 0}{\lim } \left[\frac{\mathrm{x}+2-2}{\mathrm{x}}\right]\left[\frac{1}{{\left(\mathrm{x}+2\right)}^{\frac{2}{3}}+2^{\frac{2}{3}}+{\left(\mathrm{x}+2\right)}^{\frac{1}{3}}.2^{\frac{1}{3}}}\right] \\ =\underset{\mathrm{x}\to 0}{\lim } \left[\frac{\mathrm{x}}{\mathrm{x}}\right]\left[\frac{1}{{\left(\mathrm{x}+2\right)}^{\frac{2}{3}}+2^{\frac{2}{3}}+{\left(\mathrm{x}+2\right)}^{\frac{1}{3}}.2^{\frac{1}{3}}}\right] \\ =\underset{\mathrm{x}\to 0}{\lim } \left[\frac{1}{{\left(\mathrm{x}+2\right)}^{\frac{2}{3}}+2^{\frac{2}{3}}+{\left(\mathrm{x}+2\right)}^{\frac{1}{3}}.2^{\frac{1}{3}}}\right] \\ \mathrm{Applying} \mathrm{limit}, \mathrm{we} \mathrm{get}: \\ \mathrm{L}= \left[\frac{1}{{\left(2\right)}^{\frac{2}{3}}+2^{\frac{2}{3}}+{\left(2\right)}^{\frac{1}{3}}.2^{\frac{1}{3}}}\right]=\left[\frac{1}{2^{\frac{2}{3}}+2^{\frac{2}{3}}+2^{\frac{2}{3}}}\right]=\frac{1}{3\left(2^{\frac{2}{3}}\right)} \\ \Rightarrow \boxed{\underset{\mathbf{x}\mathbf{\to }\mathbf{0}}{\mathbf{lim}}\mathbf{ }\frac{{\left(\mathbf{x}\mathbf{+}\mathbf{2}\right)}^{{\displaystyle \frac{1}{3}}}\mathbf{-}{\mathbf{2}}^{{\displaystyle \frac{1}{3}}}}{\mathbf{x}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}\left({\mathbf{2}}^{\frac{\mathbf{2}}{\mathbf{3}}}\right)}} \end{gathered}$$

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