Evaluate the following limit: `(lim)_(x->a)(x^(2//3)-a^(2//3))/(x^(3//4)-a^(3//4))`

Explanation

To evaluate the limit

\[ L = \lim_{x \to a} \frac{x^{2/3} - a^{2/3}}{x^{3/4} - a^{3/4}}, \]

we first notice that if we directly substitute \(x = a\), both the numerator and denominator become zero, leading to the indeterminate form \( \frac{0}{0} \). To resolve this, we can apply L'Hôpital's Rule, which states that if we have an indeterminate form \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), we can take the derivative of the numerator and the derivative of the denominator separately.

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