To compute the limit \( \lim_{h \to 0} \frac{f(5 + h) – f(5)}{h} \), given the function \( f(x) = 12x \), we first need to substitute the values into the limit expression.
1. **Evaluate \( f(5) \):**
Let’s calculate \( f(5) \):
- \( f(5) = 12 \times 5 = 60 \)
2. **Evaluate \( f(5 + h) \):**
Next, we need to find \( f(5 + h) \):
- \( f(5 + h) = 12 \times (5 + h) \)
- \( = 60 + 12h \)
3. **Set up the limit:**
Now, we can substitute these values into our limit expression:
- \( \frac{f(5 + h) – f(5)}{h} = \frac{(60 + 12h) – 60}{h} \)
- \( = \frac{12h}{h} \)
- \( = 12 \) (for \( h \neq 0 \) )
4. **Finding the limit:**
As we take the limit of the expression as \( h \) approaches 0, the result remains constant:
- \( \lim_{h \to 0} 12 = 12 \)
Therefore, the limit is:
\( 12 \)
In conclusion, we’ve calculated that:
\( \lim_{h \to 0} \frac{f(5 + h) – f(5)}{h} = 12 \)