(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

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.

3. 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?

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# Homework Help: Calculus I - Finding a limit

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