Question

How do you find the limit of `R(x) = (x - 1)/(x² - 1)` as x approaches `1`?

Answer 1

`1/2`

Explanation:

Note first that the denominator factors into `(x-1)(x+1)`, using the difference of squares formula. Now `R(x)` can be written like so:
`R(x)=(x-1)/((x-1)(x+1))`

We can cancel the two `(x-1)` terms to get:
`R(x)=cancel(x-1)/((cancel(x-1))(x+1))=1/(x+1)`

Now we can find `lim_(x->1)R(x)` by simply plugging `1` in for `x`:
`lim_(x->1)R(x)=lim_(x->1)(1/(x+1))=1/((1)+1)=1/2`