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The given limit problem is:

\[
\lim_{{x \to 4}} \frac{{-4x}}{{\sqrt{5x - 4}}}
\]

### Step-by-step solution:

1. **Substitute \( x = 4 \)** directly into the expression:

   \[
   \frac{{-4(4)}}{{\sqrt{5(4) - 4}}} = \frac{{-16}}{{\sqrt{20 - 4}}} = \frac{{-16}}{{\sqrt{16}}} = \frac{{-16}}{{4}} = -4
   \]

So, the limit is:

\[
\lim_{{x \to 4}} \frac{{-4x}}{{\sqrt{5x - 4}}} = -4
\]

Therefore, the correct answer is **A.** \( -4 \).

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Do you want further clarification or additional details on this solution?

Here are 5 related questions to further your understanding:
1. What happens if you substitute a different value of \( x \) in the same expression?
2. How do you handle limits that approach a number where the denominator becomes 0?
3. What if the expression under the square root was negative, how would the limit change?
4. How can you apply L'Hopital's Rule to limits involving indeterminate forms?
5. What is the difference between one-sided and two-sided limits?

**Tip:** When directly substituting a value in a limit, always check for indeterminate forms like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), which may require additional methods like factoring or L'Hopital's Rule.

Find the limit lim (x -> 4) of (-4x) / sqrt(5x - 4).

This page explains how to calculate the limit of (-4x) / sqrt(5x - 4) as x approaches 4. Learn step-by-step how to apply direct substitution and simplify the expression. Ideal for high school students studying limits and algebra.

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