Question

Question #b8966

Answer 1

`lim_(xrarr1)(1/x^2-1)/(x-1)=-2`

Explanation:

`L=lim_(xrarr1)(1/x^2-1)/(x-1)`

Multiply the numerator and denominator by `x^2` to clear the fraction in the numerator:

`L=lim_(xrarr1)(x^2(1/x^2-1))/(x^2(x-1))=lim_(xrarr1)(1-x^2)/(x^2(x-1))`

Factor the numerator as a difference of squares:

`L=lim_(xrarr1)((1+x)(1-x))/(x^2(x-1))`

Note that `1-x=-(x-1)`:

`L=lim_(xrarr1)(-(x+1)(x-1))/(x^2(x-1))=lim_(xrarr1)(-(x+1))/x^2=(-(1+1))/1^2=-2`