1. **State the problem:** Find the limit $\lim_{n \to 0} \frac{(4+n)^2 - 16}{n}$. 2. **Recall the formula:** This is a difference quotient form, often used to find derivatives. 3. **Expand the numerator:** $$ (4+n)^2 - 16 = (16 + 8n + n^2) - 16 = 8n + n^2 $$ 4. **Rewrite the limit:** $$ \lim_{n \to 0} \frac{8n + n^2}{n} $$ 5. **Simplify by canceling $n$:** $$ \lim_{n \to 0} \frac{\cancel{n}(8 + n)}{\cancel{n}} = \lim_{n \to 0} (8 + n) $$ 6. **Evaluate the limit by substituting $n=0$:** $$ 8 + 0 = 8 $$ **Final answer:** $8$