Question

How do you find the limit of `(|x| -1)/((x+1)x)` as x approaches -1?

Answer 1

`lim_{x->-1} (|x| -1)/((x+1)x) =1`

Explanation:

`(|x| -1)/((x+1)x) = abs(x)/x cdot 1/(x+1)-1/(x(x+1))`

for `x` near `-1` we have `abs(x)/x = -1` so, always near `-1`

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

then

`lim_{x->-1} (|x| -1)/((x+1)x) =lim_{x->-1}-1/x=1`