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