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Limit for roots:
$$\lim_{x \to a} \frac{\sqrt[n]{x} - \sqrt[n]{a}}{x - a} = \frac{1}{n} a^{\frac{1}{n} - 1}$$
Special case: $$\lim_{x \to 0} \frac{\sqrt[3]{a + x} - \sqrt[3]{a}}{x} = \frac{1}{3a^{2/3}}$$.
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Limit for roots:
$$\lim_{x \to a} \frac{\sqrt[n]{x} - \sqrt[n]{a}}{x - a} = \frac{1}{n} a^{\frac{1}{n} - 1}$$
Special case: $$\lim_{x \to 0} \frac{\sqrt[3]{a + x} - \sqrt[3]{a}}{x} = \frac{1}{3a^{2/3}}$$.
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