Question

How do you evaluate the limit `(s-1)/(s^2-1)` as s approaches `1`?

Answer 1

`lim_(s->1) (s-1)/(s^2-1) = 1/2`

Explanation:

Note that `s^2-1 = (s-1)(s+1)`, so we can factor and simplify the given rational expression:

`(s-1)/(s^2-1) = (color(red)(cancel(color(black)(s-1))))/((color(red)(cancel(color(black)(s-1))))(s+1)) = 1/(s+1)`

with exclusion `s != 1`

The simplified rational expression describes a function identical to the original one, except that the original is undefined at the point `s=1`.

Hence:

`lim_(s->1) (s-1)/(s^2-1) = lim_(s->1) 1/(s+1) = 1/2`