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To evaluate this limit:

\[
\lim_{{x \to 1}} \frac{\sqrt{24 + x - 5x^{1/5}}}{x^2 - 6x + 5}
\]

### Step-by-Step Solution

1. **Check the Limit by Direct Substitution**:
   Substitute \( x = 1 \) directly to check if the expression is defined.

   - **Numerator**: 
     \[
     \sqrt{24 + 1 - 5 \cdot 1^{1/5}} = \sqrt{24 + 1 - 5} = \sqrt{20}
     \]

   - **Denominator**:
     \[
     1^2 - 6 \cdot 1 + 5 = 1 - 6 + 5 = 0
     \]

   The denominator becomes zero, which suggests that direct substitution doesn’t work, and we may have an indeterminate form (non-zero in the numerator, zero in the denominator). This implies we should factorize or simplify.

2. **Factorize the Denominator**:
   The denominator \( x^2 - 6x + 5 \) can be factored:
   \[
   x^2 - 6x + 5 = (x - 1)(x - 5)
   \]

   Thus, the limit becomes:
   \[
   \lim_{{x \to 1}} \frac{\sqrt{24 + x - 5x^{1/5}}}{(x - 1)(x - 5)}
   \]

3. **Apply L'Hôpital's Rule**:
   Since we have an indeterminate form, we can apply L'Hôpital's Rule, which involves taking derivatives of the numerator and the denominator until the limit is determined.

4. **Further Steps**:
   We would continue by differentiating the numerator and denominator, which will involve the chain rule and possibly implicit differentiation due to the fractional exponent in the numerator.

Would you like me to continue with the differentiation step-by-step, or do you have specific questions about any part so far?

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Here are some related questions for further exploration:

1. How is L'Hôpital's Rule applied to limits involving radicals and fractional exponents?
2. Why is it essential to check for indeterminate forms before applying L'Hôpital's Rule?
3. Can limits with radicals sometimes be simplified using conjugates?
4. How can we handle fractional exponents when differentiating?
5. What other methods are available if L'Hôpital's Rule fails to resolve a limit?

**Tip**: Always try factoring or simplifying expressions before resorting to L'Hôpital's Rule, as simpler techniques can sometimes yield the answer directly.

Evaluate the limit \( \lim_{{x \to 1}} \frac{\sqrt{24 + x - 5x^{1/5}}}{x^2 - 6x + 5} \).

Learn how to evaluate the limit \( \lim_{{x \to 1}} \frac{\sqrt{24 + x - 5x^{1/5}}}{x^2 - 6x + 5} \) by using algebraic techniques and L'Hôpital's Rule. This calculus problem covers key concepts such as indeterminate forms, factoring, and differentiation with radicals. Suitable for college-level calculus students.

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