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We are asked to evaluate the following limit as \( x \to 0 \):

\[
\lim_{x \to 0} \frac{\sqrt{x} - \sqrt[4]{x}}{x - \sqrt[3]{x} + \sqrt[5]{x}}.
\]

### Step-by-step solution:

1. **Rewrite the terms using exponents**:
   - \(\sqrt{x} = x^{1/2}\)
   - \(\sqrt[4]{x} = x^{1/4}\)
   - \(\sqrt[3]{x} = x^{1/3}\)
   - \(\sqrt[5]{x} = x^{1/5}\)

Thus, the expression becomes:

\[
\lim_{x \to 0} \frac{x^{1/2} - x^{1/4}}{x - x^{1/3} + x^{1/5}}.
\]

2. **Factor out the smallest powers of \(x\)**:
   - In the numerator, the smallest power is \(x^{1/4}\), so factor out \(x^{1/4}\) from the terms:
   
     \[
     x^{1/4}(x^{1/4} - 1).
     \]
     
   - In the denominator, the smallest power is \(x^{1/5}\), so factor out \(x^{1/5}\) from the terms:
   
     \[
     x^{1/5}\left(x^{4/5} - x^{2/15} + 1\right).
     \]

Now, the limit expression becomes:

\[
\lim_{x \to 0} \frac{x^{1/4}(x^{1/4} - 1)}{x^{1/5}\left(x^{4/5} - x^{2/15} + 1\right)}.
\]

3. **Simplify the powers of \(x\)**:
   When dividing powers of \(x\), subtract the exponents. In this case, we subtract \(1/5\) from \(1/4\) in the overall factor:

\[
x^{1/4 - 1/5} = x^{5/20 - 4/20} = x^{1/20}.
\]

Thus, the expression simplifies to:

\[
\lim_{x \to 0} x^{1/20} \cdot \frac{(x^{1/4} - 1)}{x^{4/5} - x^{2/15} + 1}.
\]

4. **Evaluate the limit**:
   As \( x \to 0 \):
   - \( x^{1/20} \to 0 \), and hence the entire expression tends to 0.
   
Thus, the value of the limit is:

\[
\boxed{0}.
\]

Would you like further details on any specific step or have more questions?

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Here are five related questions you can try next:
1. What is the limit of \( \lim_{x \to 0} \frac{x^{1/3} - x^{1/4}}{x - x^{1/5}} \)?
2. How does factoring help in simplifying limits involving fractional powers?
3. How do you evaluate limits involving indeterminate forms?
4. What is the general approach for limits of the form \( \frac{f(x)}{g(x)} \) as \(x \to 0\)?
5. Can you apply L'Hôpital's Rule to solve this problem?

**Tip**: When dealing with limits involving roots or fractional powers, always check if you can simplify by factoring out the smallest power.

lim_{x->0} ((sqrt(x) - 4th root(x)) / (x - 3rd root(x) + 5th root(x)))

Learn how to evaluate the limit of a complex expression involving square roots, cube roots, and fifth roots as x approaches 0. This problem explores key calculus concepts such as factoring and limit evaluation, suitable for advanced high school students.

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