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The given limit expression is:

\[
\lim_{h \to 0} \frac{3h^2}{4h + 3h^2}
\]

Let's solve this algebraically:

### Step 1: Factor the common terms (if possible)
We can try simplifying the expression by factoring \(h\) in both the numerator and the denominator:

\[
\frac{3h^2}{4h + 3h^2} = \frac{3h^2}{h(4 + 3h)}
\]

Now, cancel out the common factor of \(h\) (but only if \(h \neq 0\)):

\[
\frac{3h^2}{h(4 + 3h)} = \frac{3h}{4 + 3h}
\]

### Step 2: Evaluate the limit as \(h \to 0\)
Now, substitute \(h = 0\) into the simplified expression:

\[
\lim_{h \to 0} \frac{3h}{4 + 3h} = \frac{3(0)}{4 + 3(0)} = \frac{0}{4} = 0
\]

### Conclusion:
The limit exists and is a finite number, \( \boxed{0} \).

So, the correct option is:

- **The limit does exist and is a finite number.**

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Would you like further details or have any questions? Here are some related questions for you to consider:

1. How do we handle indeterminate forms in limits?
2. What happens if the numerator and denominator of a limit both tend to zero?
3. Can L'Hôpital's Rule be applied to this problem? Why or why not?
4. How do we determine when a limit does not exist?
5. What other methods can we use to solve complex limit problems?

**Tip:** Always try simplifying the expression before plugging in the limit value. It often prevents indeterminate forms.

Calculate the limit algebraically: lim(h → 0) [3h^2 / (4h + 3h^2)].

Learn how to find the limit of a rational function as h approaches 0. This solution simplifies the expression to find the limit, highlighting key algebraic techniques. Suitable for high school or early college calculus students.

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