Evaluate
`Lim_(x to a ) (x^(3/5) -a^(3/5))/(x^(1/3)-a^(1/3))`

Answer

To evaluate the limit

\[ \lim_{x \to a} \frac{x^{3/5} - a^{3/5}}{x^{1/3} - a^{1/3}}, \]

we start by substituting \(x = a\):

1. **Substitution**:
\[ \text{If } x = a, \text{ then } x^{3/5} - a^{3/5} = 0 \text{ and } x^{1/3} - a^{1/3} = 0. \]
This gives us the indeterminate form \(\frac{0}{0}\).

**Hint**: When you encounter the indeterminate form \(\frac{0}{0}\), consider applying L'Hôpital's Rule.

2. **Applying L'Hôpital's Rule**:
We differentiate the numerator and the denominator separately:
\[ \text{Numerator: } \frac{d}{dx}(x^{3/5}) = \frac{3}{5}x^{-2/5}, \]
\[ \text{Denominator: } \frac{d}{dx}(x^{1/3}) = \frac{1}{3}x^{-2/3}. \]

3. **Rewrite the limit**:
Using L'Hôpital's Rule, we have:
\[ \lim_{x \to a} \frac{x^{3/5} - a^{3/5}}{x^{1/3} - a^{1/3}} = \lim_{x \to a} \frac{\frac{3}{5}x^{-2/5}}{\frac{1}{3}x^{-2/3}}. \]

4. **Simplifying the limit**:
This simplifies to:
\[ \lim_{x \to a} \frac{\frac{3}{5}}{\frac{1}{3}} \cdot x^{(-2/5) - (-2/3)} = \lim_{x \to a} \frac{3}{5} \cdot 3 \cdot x^{(-2/5 + 2/3)}. \]

5. **Finding a common denominator**:
The exponent \(-2/5 + 2/3\) can be calculated by finding a common denominator:
\[ -2/5 = -6/15 \quad \text{and} \quad 2/3 = 10/15, \]
so,
\[ -2/5 + 2/3 = \frac{-6 + 10}{15} = \frac{4}{15}. \]

6. **Final limit**:
Now we have:
\[ \lim_{x \to a} \frac{9}{5} x^{4/15}. \]
Substituting \(x = a\):
\[ = \frac{9}{5} a^{4/15}. \]

Thus, the final answer is:

\[ \frac{9}{5} a^{4/15}. \]