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

1 Answer
Jan 7, 2016

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