Evaluate the following limits: _ x 7 [ ( [3] x - [3] 7 )( [3] x + [3] 7 ) x-7 ]

_ x 7 [ ( [3] x - [3] 7 )( [3] x + [3] 7 ) x-7 ] = _ x 7 [ (x^ 1 3 -7^ 1 3 ) (x^ 1 3 +7^ 1 3 ) x-7 ] = _ x 7 [ x^ 2 3 -7^ 2 3 x-7 ] [ (a-b)(a+b)=a^2-b^2 ] =2 3 (7)^ -1 3 [ _ x x^n-a^n x-a =n a^ n-1 ] =2 3 1 7^ 1 3 =2 3 [3] 7

Evaluate the following limits: _ x 7 [ ( [3] x - [3] 7 )( [3] x +…

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Question

Evaluate the following limits:
$\lim _{x \rightarrow 7}\left[\frac{(\sqrt[3]{x}-\sqrt[3]{7})(\sqrt[3]{x}+\sqrt[3]{7})}{x-7}\right]$

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Answer

$ \lim _{x \rightarrow 7}\left[\frac{(\sqrt[3]{x}-\sqrt[3]{7})(\sqrt[3]{x}+\sqrt[3]{7})}{x-7}\right]$
$=\lim _{x \rightarrow 7}\left[\frac{\left(x^{\frac{1}{3}}-7^{\frac{1}{3}}\right)\left(x^{\frac{1}{3}}+7^{\frac{1}{3}}\right)}{x-7}\right]$
$=\lim _{x \rightarrow 7}\left[\frac{x^{\frac{2}{3}}-7^{\frac{2}{3}}}{x-7}\right] \quad \ldots\left[\because(\mathrm{a}-\mathrm{b})(\mathrm{a}+\mathrm{b})=\mathrm{a}^2-\mathrm{b}^2\right]$
$=\frac{2}{3}(7)^{\frac{-1}{3}} \quad \ldots\left[\because \lim _{x \rightarrow \infty} \frac{x^n-\mathrm{a}^n}{x-\mathrm{a}}=\mathrm{n} \cdot \mathrm{a}^{n-1}\right]$
$=\frac{2}{3} \cdot \frac{1}{7^{\frac{1}{3}}}$
$=\frac{2}{3 \sqrt[3]{7}} $

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