Question

What is `lim_(xrarr-1) (1/x-1)/(x-1)`?

Answer 1

`lim_(xrarr-1) (1/x-1)/(x-1)=1`

Explanation:

`lim_(xrarr-1) (1/x-1)/(x-1) = (lim_(xrarr-1) (1/x-1))/(lim_(xrarr-1)(x-1))` Provided that both limits exist and the denominator doses not go to `0`.

So, consider the limits separately.

`lim_(xrarr-1) (1/x-1) = lim_(xrarr-1) 1/x-lim_(xrarr-1) 1`

`= 1/-1-1=-1-1=-2`

So the numerator limit exists and equals `-2`

`lim_(xrarr-1)(x-1) = -1-1=-2`

So the denominator limit exists and equals `-2`

Therefore

`lim_(xrarr-1) (1/x-1)/(x-1)= (-2)/(-2) =1`