Question

What is `lim_(xrarr1) (x^2-1)/(x^3-1)`?

Answer 1

`lim_(xrarr1)(x^2-1)/(x^3-1)=color(red)(2/3)`

Explanation:

Factoring identities you should know:
`color(white)("XXX")color(blue)((a^2-b^2)=(a-b)(a+b))`
`color(white)("XXX")color(green)((a^3-b^3)=(a-b)(a^2+ab+b^2))`
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`lim_(xrarr1)(color(blue)(x^2-1))/(color(green)(x^3-1))`

`color(white)("XXX")=lim_(xrarr1)(color(blue)(cancel((x-1))(x+1)))/color(green)(cancel((x-1))(x^2+x+1))`

`color(white)("XXX")=(1+1)/(1+1+1)`

`color(white)("XXX")=2/3`