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  • Evaluate: lim(x→0)⁡((x+2)^(1/3)-2^(1/3))/x

Evaluate:  \mathop{\lim }_{x \rightarrow 0}\frac{ \left( x+2 \right) ^{\frac{1}{3}}-2^{\frac{1}{3}}}{x}

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Given~\mathop{\lim }_{x \rightarrow 0}\frac{ \left( x+2 \right) ^{\frac{1}{3}}-2^{\frac{1}{3}}}{x}~~ \\\\ ~

Now put x=x-2  limits change from 0 to 2      

 \\\\ =\mathop{\lim }_{y \rightarrow 2}\frac{y^{\frac{1}{3}}-2^{\frac{1}{3}}}{y-2}=\frac{1}{3} \left( 2 \right) ^{\frac{1}{3}-1}=\frac{1}{3}2^{-\frac{2}{3}}     \left[u \sin g \lim _{x \rightarrow a} \frac{x^{n}-a^{n}}{x-a}=n \cdot a^{n-1}\right]

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