Evaluate: _ x 0 (x+2)^1 3 -2^1 3 x - Vidyadip

Question

Evaluate:
$\lim\limits_{\text{x} \rightarrow 0}\frac{(\text{x}+2)^\frac{1}{3}-2^\frac{1}{3}}{\text{x}}$

✓

Answer

Given that $\lim\limits_{\text{x} \rightarrow 0}\frac{(\text{x}+2)^\frac{1}{3}-2^\frac{1}{3}}{\text{x}}$
Put x + 2 = y
⇒ x= y - 2
$=\lim\limits_{\text{y-2} \rightarrow 0}\frac{\text{y}^\frac{1}{3}-2^\frac{1}{3}}{\text{y}-2}$
$=\lim\limits_{\text{y} \rightarrow 2}\frac{\text{y}^\frac{1}{3}-2^\frac{1}{3}}{\text{y}-2}$
$=\frac{1}{3}.(2)^{\frac{1}{3}-1}=\frac{1}{3}.2^{\frac{-2}{3}}$
Hence, the answer is $\frac{1}{3}.(2)^{\frac{-2}{3}}.$

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