To solve the given limit problem:

$\lim_{h \to 0} \frac{(15 + h)^2 - 225}{h}$

Step 1: Expand the numerator

Expand $(15 + h)^2$: $(15 + h)^2 = 15^2 + 2(15)(h) + h^2 = 225 + 30h + h^2$

Substitute this into the numerator: $(15 + h)^2 - 225 = 225 + 30h + h^2 - 225 = 30h + h^2$

Now the expression becomes: $\lim_{h \to 0} \frac{30h + h^2}{h}$

Step 2: Simplify the fraction

Factor $h$ out of the numerator: $\frac{30h + h^2}{h} = \frac{h(30 + h)}{h}$

Cancel $h$ (for $h \neq 0$): $30 + h$

Step 3: Apply the limit

As $h \to 0$: $30 + h \to 30$

Thus, the limit is: $\boxed{30}$

Let me know if you'd like a detailed explanation or further assistance!

Related Questions:

1. What is the definition of a derivative, and how does it relate to this problem?
2. How can you verify this result using a numerical approach?
3. What happens if the numerator is cubic instead of quadratic?
4. How does this relate to finding the slope of a tangent line to a curve?
5. How would this limit change if $15$ were replaced by another number?

Tip:

When solving limits, always simplify the numerator and denominator before applying the limit to avoid indeterminate forms.