Question

How do you proof the limit of (x+1)/(x-1) is negative infinity as x approaches 1−?

`lim_(x->1^-) (x+1)/(x-1) = - oo`
How to proof that with definition?

Answer 1

Please see below.

Explanation:

To find `lim_(x->1^-) (x+1)/(x-1)` as `x` approaches `1` from left hand side, assume `x=1-h` and see how function behaves as `h->0^+`. Note that though `h->0^+`, as `h=1-x`, this means `(1-x)->0^+` or `x->1^-`.

Then `lim_(x->1^-)(x+1)/(x-1)`

= `lim_(h->0^+)(1-h+1)/(1-h-1)`

= `lim_(h->0^+)(2-h)/(-h)`

= `lim_(h->0^+)-(2-h)/h`

= `-oo`