having difficulty with doing limit proof for x^2 - 3x

[xXAceFireXx]

I understand the easy limit question, like finding lim(x->3) (5x+2) = 17. These are pretty straightforward. However, I'm having trouble doing ones such as this one:

lim(x->2) (x^2 - 3x) = -2

I know it requires the use of min(...,...) (I know the much from my notes), but I don't really know how to apply it.

Also, if anybody knows of a website that gives a good lesson on doing limit proofs, I would greatly appreciate a link. Thank you!

 

[galactus]

This one is as easy as the other.

With this, all you have to do is sub in x=2

\(\displaystyle (2)^{2}-3(2)={-}2\)


\(\displaystyle \L\\\lim_{x\to\2}(x^{2}-3x)=\lim_{x\to\2}x^{2}-\lim_{x\to\2}{3x}=\lim_{x\to\2}x^{2}-3\lim_{x\to\2}{x}=(2)^{2}-3(2)={-}2\)

 

[xXAceFireXx]

While you have evluated the limit using limit laws, this evaluation doesn't prove the limit. I apologize; I should have been more specific: I need to prove the limit using an epsilon-delta limit proof.

Thank you.

 

[pka]

I need to prove the limit using an epsilon-delta limit proof.

You are up against one of the most difficult problems for getting help on the web. This topic is so abstract as to need face-to-face instruction. I don’t think it possible to learn even from a self-help book.

You have the definition memorized. RIGHT? In not, start there; be able to say it in your sleep.

\(\displaystyle \begin{array}{rcl}
\left| {f(x) - f(2)} \right| & = & \left| {\left( {x^2 - 3x} \right) - \left( { - 2} \right)} \right| \\
& = & \left| {x^2 - 3x + 2} \right| \\
& = & \left| {\left( {x - 1} \right)\left( {x - 2} \right)} \right| \\
& = & \left| {\left( {x - 1} \right)} \right|\left| {\left( {x - 2} \right)} \right| \\
\end{array}.\)

By making delta less that 1 you have \(\displaystyle \left| {\left( {x - 1} \right)} \right| < 2.\)
So take epsilon divided by 2.

I did tell you that you really do need face-to-face instruction.

 

[galactus]

There are lots of Applets if you want to check them out.

Here's one:

http://www.slu.edu/classes/maymk/Applet ... Delta.html