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  4. The value of displaystyle limx → 1(x1/4-x1/5/x3-1) is

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Q. The value of $\displaystyle \lim_{x \to 1}$$\frac{x^{1/4}-x^{1/5}}{x^{3}-1}$ is

Limits and Derivatives

Solution:

$\displaystyle \lim_{x \to 1}$$\frac{x^{1/4}-x^{1/5}}{x^{3}-1}$
$=\displaystyle \lim_{x \to 1}$$\frac{\left(x^{1/4}-1\right)-\left(x^{1/5}-1\right)}{x^{3}-1}$
$=\displaystyle \lim_{x \to 1}$$\left(\frac{\frac{x^{1/4}-1^{1/4}}{x-1}}{\frac{x^{3}-1^{3}}{x-1}}-\frac{\frac{x^{1/5}-x^{1/5}}{x-1}}{\frac{x^{3}-1^{3}}{x-1}}\right)$
$=\frac{1}{12}-\frac{1}{15}=\frac{1}{60}$