Question

consider the following function.
$f(x) = 6x - 9$
find the limit.
$lim_{delta x \to 0} \frac{f(x + delta x) - f(x)}{delta x}$

Explanation:

Step1: Substitute $f(x+\Delta x)$ and $f(x)$

First, calculate $f(x+\Delta x) = 6(x+\Delta x) - 9 = 6x + 6\Delta x - 9$, then substitute into the limit:
$$\lim_{\Delta x \to 0} \frac{(6x + 6\Delta x - 9) - (6x - 9)}{\Delta x}$$

Step2: Simplify the numerator

Combine like terms in the numerator:
$$\lim_{\Delta x \to 0} \frac{6x + 6\Delta x - 9 - 6x + 9}{\Delta x} = \lim_{\Delta x \to 0} \frac{6\Delta x}{\Delta x}$$

Step3: Cancel common factor

Cancel $\Delta x$ from numerator and denominator (valid since $\Delta x
eq 0$ as it approaches 0):
$$\lim_{\Delta x \to 0} 6$$

Step4: Evaluate the limit

The limit of a constant is the constant itself.

Answer:

6