Thread: How do I prove this limit using the δ , ε method of proof?

1. How do I prove this limit using the δ , ε method of proof?

lim x->2 (2x + 3) / (x - 1)

using the δ , ε method of proof?

I know that:

if 0 < | x - 2 | < δ then | ( (2x + 3) / (x - 1) ) - 7 | < ε

but i'm not quite sure what to do after that

Thanks!

2. $\lim_{x \to 2} \frac{2x + 3}{x-1} = 7$

By definition:
$\forall \epsilon > 0, \ \exists \delta > 0$ such that whenever $0 < |x - 2| < \delta$ then it follows that $\left| \frac{2x+3}{x - 1} - 7 \right| < \epsilon$

In these questions, you always fudge the last expression to get $|x - 2| < \text{something}$ in terms of $\epsilon$ in which you set $\delta$ to. So:

$\begin{array}{rcl} \left| \displaystyle \frac{2x + 3}{x-1} - \frac{7(x-1)}{x-1} \right| & < & \epsilon \qquad \text{Found common denominator}\\ & & \\ \left| \displaystyle \frac{2x + 3 - 7x + 7}{x-1} \right| & < & \epsilon \\ & & \\ \left| \displaystyle \frac{-5x + 10}{x - 1} \right| & < & \epsilon \\ & & \\ |-5| \displaystyle \frac{|x-2|}{{\color{red}|x-1|}} & < & \epsilon \end{array}$

So the problem is the $|x - 1|$. To deal with this, we can assume that we're dealing with relatively small $\delta$ values so we can say $\delta \leq \frac{1}{2}$ (Normally we use $\delta \leq 1$ but we run into a bit of a problem so we choose a different value). So:
$|x - 2| < \delta \leq \frac{1}{2} \ \ \Rightarrow \ \ -\frac{1}{2} \ < \ x - 2 \ < \ \frac{1}{2}$

Adding +1 to all 3 sides we get: $\frac{1}{2} < x - 1 < \frac{3}{2} \ \Rightarrow {\color{red}2 > \frac{1}{|x - 1|}} > \frac{2}{3}$

Can you finish off?

3. Can you finish it please?

I understand what you've done so far, but don't know what to do next. :P

4. It isn't that big a leap.

You want to get $|x - 2| < \text{Something}$. Then we'll let $\delta = \text{Something}$ and we're done.

We have: $5\frac{{\color{red}1}|x-2|}{{\color{red}|x-1|}} < \epsilon$ and ${\color{red}\frac{1}{|x-1|} < 2}$.

So we know that: $5\frac{|x-2|}{|x-1|} < \hdots$ ??? Think!

5. ε / 10 = δ?

6. Originally Posted by coldfire
or $\delta = \frac{2\epsilon}{5}?$

I'm so confused
we have $5 \cdot \frac {|x - 2|}{|x - 1|} < \epsilon$ with $\frac 1{|x - 1|} < 2$

thus, $5 \cdot \frac {|x - 2|}{|x - 1|} < 5 \cdot 2 |x - 2| = 10|x - 2| < \epsilon$

so ...

7. so it's $\delta = \frac{\epsilon}{10}$?

8. Originally Posted by coldfire
so it's $\delta = \frac{\epsilon}{10}$?
yes

9. ok thanks guys

for the answer, do I write:

$\delta = \frac{\epsilon}{10}$ if $\frac{1}{|x-1|} < 2$?

or if not, what do I put after if?

10. Originally Posted by coldfire
ok thanks guys

for the answer, do I write:

$\delta = \frac{\epsilon}{10}$ if $\frac{1}{|x-1|} < 2$?

or if not, what do I put after if?
no. remember, you are trying to prove that 7 is thew limit. restate the definition of what the limit is and say something along the lines of, "choose $\delta = \frac {\epsilon}{10}$, then blah bla blah is less than $\epsilon$, as desired"

11. Actually it's $\delta = \text{min} \left( \frac{1}{2}, \frac{\epsilon}{10}\right)$ because you have 2 inequalities to be satisfied: $|x - 2| < \frac{1}{2}$ and $|x-2| < \frac{\epsilon}{2}$.

The lower of the two will satisfy both, hence our choice of $\delta$

12. Originally Posted by o_O
Actually it's $\delta = \text{min} \left( \frac{1}{2}, \frac{\epsilon}{10}\right)$ because you have 2 inequalities to be satisfied: $|x - 2| < \frac{1}{2}$ and $|x-2| < \frac{\epsilon}{2}$.

The lower of the two will satisfy both, hence our choice of $\delta$
yep, i forgot that. we were concentrating so hard on solving that inequality

13. Originally Posted by o_O
$\lim_{x \to 2} \frac{2x + 3}{x-1} = 7$

(Normally we use $\delta \leq 1$ but we run into a bit of a problem so we choose a different value).
Sorry to bump this problem but o_O, could you explain why you chose 1/2 instead of 1, I used 1 and I thought I had achieved the same result. I read your explanation but in which cases should I consider a value other than 1? I'm having a hard time trying to grasp this

14. The thing is you don't have to work with $\delta \leq 1$. Generally, this will work but sometimes it just messes up the algebra so just pick another small value for $\delta$.

If we used $\delta \leq 1$, we would have: $|x - 2| < \delta = 1 \ \Rightarrow \ -1 \ < \ x-2 \ < \ 1$

Adding +1 to both sides we get: $0 \ < \ |x - 1| \ < \ 2$

Now the point of doing all this is to get: $\frac{1}{|x-1|} < a$ where $a$ is some constant.

As we can see, if we tried to take the reciprical of all three sides (and thereby switching the inequality around), we would have:
$\frac{1}{2} < \frac{1}{|x-1|} < "\frac{1}{0}"$

which means we don't have an upper bound for $\frac{1}{|x-1|}$.

So I picked another value for $\delta$ that won't cause this problem.

15. Thank you for your explanation o_0!