Question

What is the limit as `x -> 0` of `((1/(x-1) + 1 )/ x)`?

Answer 1

graph{((1/(x-1))+1)/x [-2.737, 2.737, -1.367, 1.37]}

`lim_(xrarr0)((1/(x-1)+1)/x)=-1`

Explanation:

`lim_(xrarr0)((1/(x-1)+1)/x)=lim_(xrarr0)((cancel(1)+xcancel(-1))/(x-1))/x=`

`lim_(xrarr0)((x)/(x-1))/x=lim_(xrarr0)1/cancel(x)*cancel(x)/(x-1)=`
`=lim_(xrarr0)1/(x-1)`

`1/(x-1)` is continuos in `x=0`

then:

`=lim_(xrarr0)1/(x-1)=1/(0-1)=1/-1=-1`