# Homework Help: Calculus I - Finding a limit

1. Sep 14, 2010

1. The problem statement, all variables and given/known data

Consider the following function:

$$f(x) = 4x^2 - 8x.$$

Find the limit.

$$\lim_{{\Delta}x\rightarrow 0} \frac{f(x+{\Delta}x)-f(x)}{{\Delta}x}$$

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 $${\Delta}x$$ in the denominator.

Here's what I did first:

$$\frac{4(x+{\Delta}x)^2-8(x+{\Delta}x)}{{\Delta}x}$$

After expanding:

$$\frac{4(x^2+2x{\Delta}x+{\Delta}x^2)-8(x+{\Delta}x)}{{\Delta}x}$$

After distributing:

$$\frac{4x^2+8x{\Delta}x+4{\Delta}x^2-8x-8{\Delta}x}{{\Delta}x}$$

Would my next step be?:

$$\frac{(4{\Delta}x^2+8x{\Delta}x-8{\Delta}x)+(4x^2-8x)}{{\Delta}x}$$

...so that I could pull out the $${\Delta}x$$ and cancel it?

2. Sep 14, 2010

### Dick

You forgot to subtract the 'f(x)' that's in the numerator of your difference quotient. That's what cancels the stuff that doesn't have a delta-x in it.

3. Sep 14, 2010

### curiousphoton

The second group (4x^2-8x) in the numerator does not have a {\Delta}x so you can't completely get rid of the {\Delta}x. Do you have any other attemps?

4. Sep 14, 2010

What an embarrassingly careless mistake. Thank you, Dick. I've obtained the correct expression for the limit.

My initial difference quotient should've been:

$$\frac{4(x+{\Delta}x)^2-8(x+{\Delta}x)-(4x^2-8x)}{{\Delta}x}$$

Thanks for the help!