Let's start by analyzing the function:
[tex]\[ f(x) = 4x^2 - 6x \][/tex]
We are asked to find the limit:
[tex]\[ \lim_{{\Delta x \to 0}} \frac{f(x + \Delta x) - f(x)}{\Delta x} \][/tex]
First, we need to compute [tex]\( f(x + \Delta x) \)[/tex]. Substituting [tex]\( x + \Delta x \)[/tex] into [tex]\( f \)[/tex], we get:
[tex]\[ f(x + \Delta x) = 4(x + \Delta x)^2 - 6(x + \Delta x) \][/tex]
Expanding this expression:
[tex]\[ f(x + \Delta x) = 4(x^2 + 2x \Delta x + (\Delta x)^2) - 6(x + \Delta x) \][/tex]
[tex]\[ = 4x^2 + 8x \Delta x + 4(\Delta x)^2 - 6x - 6\Delta x \][/tex]
Next, we need to find the difference [tex]\( f(x + \Delta x) - f(x) \)[/tex]:
[tex]\[ f(x + \Delta x) - f(x) = (4x^2 + 8x \Delta x + 4(\Delta x)^2 - 6x - 6\Delta x) - (4x^2 - 6x) \][/tex]
[tex]\[ = 4x^2 + 8x \Delta x + 4(\Delta x)^2 - 6x - 6\Delta x - 4x^2 + 6x \][/tex]
Simplifying this expression:
[tex]\[ f(x + \Delta x) - f(x) = 8x \Delta x + 4(\Delta x)^2 - 6\Delta x \][/tex]
Now, we must divide this by [tex]\(\Delta x\)[/tex]:
[tex]\[ \frac{f(x + \Delta x) - f(x)}{\Delta x} = \frac{8x \Delta x + 4(\Delta x)^2 - 6\Delta x}{\Delta x} \][/tex]
We can factor out a [tex]\(\Delta x\)[/tex] from the numerator:
[tex]\[ = \frac{\Delta x (8x + 4\Delta x - 6)}{\Delta x} \][/tex]
Cancelling [tex]\(\Delta x\)[/tex] from the numerator and the denominator (assuming [tex]\(\Delta x \neq 0\)[/tex]):
[tex]\[ = 8x + 4\Delta x - 6 \][/tex]
Finally, we take the limit as [tex]\(\Delta x\)[/tex] approaches 0:
[tex]\[ \lim_{{\Delta x \to 0}} (8x + 4\Delta x - 6) = 8x - 6 \][/tex]
Therefore, the limit is:
[tex]\[ \lim_{{\Delta x \to 0}} \frac{f(x+\Delta x)-f(x)}{\Delta x} = 8x - 6 \][/tex]
