Question

Evaluate
$\rm \displaystyle \lim_{x\to 3}\ \frac{\sqrt[4]{x} -\sqrt[4]{3}}{\sqrt[3]{x}-\sqrt[3]{3}}$

Answer 1

Solution :

$\rm \displaystyle \lim_{x\to 3}\ \frac{\sqrt[4]{x} -\sqrt[4]{3}}{\sqrt[3]{x}-\sqrt[3]{3}}$

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We know that ,

$\rm \displaystyle \lim_{x\to a}\ \frac{x^m -a^m}{x^n-a^n}$

= (m/n )× $a^{(m-n)}$

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= (1/4)/(1/3)×$3^{(1/4-1/3)}$

= 3/4 × $3^{(-1/12)}$

= 1/4×$3^{(1-1/12)}$

= 1/4× $3^{(11/12)}$

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