# Math Help - Can someone explain why this line of reasoning concerning ordered relations is right?

1. ## Can someone explain why this line of reasoning concerning ordered relations is right?

Choose δ ∈ (0,1) such that 1 - δ < x < 1 implies 3/M < 2x2 - 3x + 1 < 0; i.e., M/3 > 1/(2x2 - 3x + 1). Notice that 0 < x < 1 also implies 2 < x + 2 < 3. It follows that f(x) = (x+2)/(2x2 - 3x + 1) < M for all 1 - δ < x < 1.

I do not understand the last statement because it seems to say that if 0 < a < b and d < c < 0, then ad < bc. Is this true?

2. ## Re: Can someone explain why this line of reasoning concerning ordered relations is ri

Originally Posted by lm1988
Choose δ ∈ (0,1) such that 1 - δ < x < 1 implies 3/M < 2x2 - 3x + 1 < 0; i.e., M/3 > 1/(2x2 - 3x + 1). Notice that 0 < x < 1 also implies 2 < x + 2 < 3. It follows that f(x) = (x+2)/(2x2 - 3x + 1) < M for all 1 - δ < x < 1.
I do not understand the last statement because it seems to say that if 0 < a < b and d < c < 0, then ad < bc. Is this true?

Personally, I think that it is unreasonable for you to ask for explanation starting in the middle of someone's else proof.

Why not simply state the actual question and ask for help?

3. ## Re: Can someone explain why this line of reasoning concerning ordered relations is ri

Originally Posted by Plato
Personally, I think that it is unreasonable for you to ask for explanation starting in the middle of someone's else proof.

Why not simply state the actual question and ask for help?
There is no question. I am just asking what the line of reasoning is as in the original post. The only additional information that you need to know is that M < 0 and is in R. This is actually how the proof begins.

This is what I am trying to prove:
$\lim_{x \to 1-}f(x):=\frac{x+2}{2x^{2}-3x+1}=-\infty$

4. ## Re: Can someone explain why this line of reasoning concerning ordered relations is ri

Originally Posted by lm1988
There is no question. I am just asking what the line of reasoning is as in the original post. The only additional information that you need to know is that M < 0 and is in R. This is actually how the proof begins.
Again, without knowing the reason for that argument, it impossible to comment on the argument itself.

Here is the best guess I have: if
\begin{align*}\frac{3}{M} &< f <0\\ \frac{3}{fM} &> 1\\\frac{1}{fM} &>\frac{1}{3}\\ \frac{1}{f}&<\frac{M}{3} \end{align*}.

5. ## Re: Can someone explain why this line of reasoning concerning ordered relations is ri

Originally Posted by Plato
Again, without knowing the reason for that argument, it impossible to comment on the argument itself.

Here is the best guess I have: if
\begin{align*}\frac{3}{M} &< f <0\\ \frac{3}{fM} &> 1\\\frac{1}{fM} &>\frac{1}{3}\\ \frac{1}{f}&<\frac{M}{3} \end{align*}.
The proof was provided in a textbook, so I am assuming that the author intended for the reader to understand the reason for that argument.