- #1
Dembadon
Gold Member
- 659
- 89
Homework Statement
Consider the following function:
[tex]f(x) = 4x^2 - 8x.[/tex]
Find the limit.
[tex]
\lim_{{\Delta}x\rightarrow 0} \frac{f(x+{\Delta}x)-f(x)}{{\Delta}x}
[/tex]
Given: The limit exists.
The Attempt at a Solution
Since the limit exists, I know that I need to do some algebraic manipulations that will enable me to cancel the [tex]{\Delta}x[/tex] in the denominator.
Here's what I did first:
[tex]
\frac{4(x+{\Delta}x)^2-8(x+{\Delta}x)}{{\Delta}x}
[/tex]
After expanding:
[tex]
\frac{4(x^2+2x{\Delta}x+{\Delta}x^2)-8(x+{\Delta}x)}{{\Delta}x}
[/tex]
After distributing:
[tex]
\frac{4x^2+8x{\Delta}x+4{\Delta}x^2-8x-8{\Delta}x}{{\Delta}x}
[/tex]
Would my next step be?:
[tex]
\frac{(4{\Delta}x^2+8x{\Delta}x-8{\Delta}x)+(4x^2-8x)}{{\Delta}x}
[/tex]
...so that I could pull out the [tex]{\Delta}x[/tex] and cancel it?