Epsilon-delta proof of one sided infinite limit.

[reinloch]

Homework Statement

proof this limit:
$\lim_{x\rightarrow 1^+}\frac{1}{(x-1)(x-2)}=-∞$

Homework Equations

The Attempt at a Solution

So for every $N < 0$, I need to find a $\delta > 0$ such that
$0 < x - 1 < \delta \Rightarrow \frac{1}{(x-1)(x-2)} < N$

Assuming $0 < x - 1 < 1$, I get $-1 < x - 2 < 0$, and $-\frac{1}{x-2}>1$.

Assuming $0 < x - 1 < -\frac{1}{N}$, I get $-(x-1) > \frac{1}{N}$, $-\frac{1}{x-1} < N$, and $\left(-\frac{1}{x-1}\right)\left(-\frac{1}{x-2}\right) < N\left(-\frac{1}{x-2}\right)$, but then I got stuck.

 

[tiny-tim]

welcome to pf!

hi reinloch! welcome to pf!

$\left(-\frac{1}{x-1}\right)\left(-\frac{1}{x-2}\right) < N\left(-\frac{1}{x-2}\right)$

the trick is to choose δ so that 1/(x - 2) is less than a fixed number

 

[reinloch]

Thanks. I am stuck with the right choice for $\delta$. I choose 1 and $-\frac{1}{N}$, and it didn't seem to work.

 

[tiny-tim]

choose δ so that x doesn't get too close to 2