We are tasked with finding the limit of the expression:

\[
\lim_{h \to 0} \frac{\sqrt{5h + 4} - 2}{h}
\]

### Step-by-step solution:

1. **Initial substitution**: 
   Substituting \( h = 0 \) directly into the expression gives:

   \[
   \frac{\sqrt{5(0) + 4} - 2}{0} = \frac{\sqrt{4} - 2}{0} = \frac{2 - 2}{0} = \frac{0}{0}
   \]

   This results in an indeterminate form \( \frac{0}{0} \), so we need to simplify further.

2. **Rationalizing the numerator**: 
   To handle the indeterminate form, we can multiply the expression by the conjugate of the numerator. The conjugate of \( \sqrt{5h + 4} - 2 \) is \( \sqrt{5h + 4} + 2 \). Multiplying both the numerator and denominator by this conjugate:

   \[
   \frac{\sqrt{5h + 4} - 2}{h} \cdot \frac{\sqrt{5h + 4} + 2}{\sqrt{5h + 4} + 2}
   \]

   This simplifies to:

   \[
   \frac{(\sqrt{5h + 4})^2 - 2^2}{h(\sqrt{5h + 4} + 2)} = \frac{5h + 4 - 4}{h(\sqrt{5h + 4} + 2)} = \frac{5h}{h(\sqrt{5h + 4} + 2)}
   \]

3. **Cancel the \( h \)**:
   Now, the \( h \) in the numerator and denominator cancels out:

   \[
   \frac{5}{\sqrt{5h + 4} + 2}
   \]

4. **Substitute \( h = 0 \)**:
   Now that the expression is simplified, substitute \( h = 0 \):

   \[
   \frac{5}{\sqrt{5(0) + 4} + 2} = \frac{5}{\sqrt{4} + 2} = \frac{5}{2 + 2} = \frac{5}{4}
   \]

Thus, the limit is:

\[
\lim_{h \to 0} \frac{\sqrt{5h + 4} - 2}{h} = \frac{5}{4}
\]

### Final Answer:
\[
\boxed{\frac{5}{4}}
\]

Would you like more details or have any questions?

Here are 5 related questions for practice:
1. Find \(\lim_{h \to 0} \frac{\sqrt{3h + 9} - 3}{h}\).
2. Evaluate \(\lim_{x \to 1} \frac{\sqrt{2x + 3} - 2}{x - 1}\).
3. Solve \(\lim_{x \to 0} \frac{\sqrt{x + 1} - 1}{x}\).
4. Find the derivative of \( f(h) = \sqrt{5h + 4} \) using first principles.
5. Evaluate \(\lim_{h \to 0} \frac{\sqrt{7h + 1} - 1}{h}\).

**Tip**: Rationalizing expressions often helps resolve indeterminate forms in limits involving square roots.

Learn how to solve the limit lim h -> 0 (sqrt(5h + 4) - 2)/h using rationalization to eliminate the indeterminate form 0/0. Step-by-step solution involves rationalizing the numerator and simplifying. This is suitable for Grades 11-12 and college students studying calculus and limits.

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