Question

How do you evaluate the limit of `lim (1/x-1)/(x-1)` as `x->1`?

Answer 1

`lim_(x->1)(1/x-1)/(x-1)=-1`

Explanation:

`(1/x-1)/(x-1)=((1-x)/x)/(x-1)=-(x-1)/(x(x-1))=-1/x`

Hence `lim_(x->1)(1/x-1)/(x-1)=lim_(x->1)(-1/x)=-1/1=-1`
graph{(1/x-1)/(x-1) [-5.03, 4.97, -2.4, 2.6]}