Evaluate the following limit: _ x a (x+2)^ 3 2 -(a+2)^ 3 2 x-a - Vidyadip

Question

Evaluate the following limit: $\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{(\text{x}+2)^{\frac{3}{2}}-(\text{a}+2)^{\frac{3}{2}}}{\text{x}-\text{a}}$

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Answer

$\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{(\text{x}+2)^{\frac{3}{2}}-(\text{a}+2)^{\frac{3}{2}}}{\text{x}-\text{a}}$ $=\lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{(\text{x}+2)^{\frac{3}{2}}-(\text{a}+2)^{\frac{3}{2}}}{(\text{x}+2)-(\text{a}+2)}$ Let x + 2 = y, a + 2 = b $\Rightarrow\lim\limits_{({\text{x}+2)}\rightarrow{(\text{a}+2)}}\frac{(\text{y})^\frac{3}{2}-(\text{b})^\frac{3}{2}}{(\text{y})-(\text{b})},$ $\Big[\text{Using formula}\ \lim\limits_{\text{x}\rightarrow{\text{a}}}\frac{\text{x}^{\text{n}}-\text{a}^\text{n}}{\text{x}-\text{a}}=\text{na}^{\text{n}-1}\Big]$ $=\frac{3}{2}(\text{b})^{\frac{3}{2}-1}$ $=\frac{3}{2}(\text{a}+2)^{\frac{3}{2}-1}$ $=\frac{3}{2}(\text{a}+2)^{\frac{1}{2}}$

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