Calculating Limits for a Quadratic Function

[Dembadon]

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?

 

[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.

 

[curiousphoton]

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?

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?

 

[Dembadon]

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.

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

My initial difference quotient should've been:

[tex]
\frac{4(x+{\Delta}x)^2-8(x+{\Delta}x)-(4x^2-8x)}{{\Delta}x}
[/tex]

Thanks for the help!