Question

Question #becac

Answer 1

`lim_(x->-1)(sqrt(x^2+8)-3)/(x+1)=-1/3`

Explanation:

As we are getting a `0` in the numerator and the denominator, on direct substitution, our strategy will be to rationalize the numerator by using the identity `(a+b)(a-b) = a^2-b^2`. When we have two polynomial expressions with `-1` as a root, we will be able to cancel the `x+1` factors.

`lim_(x->-1)(sqrt(x^2+8)-3)/(x+1) = lim_(x->-1)((sqrt(x^2+8)-3)(sqrt(x^2+8)+3))/((x+1)(sqrt(x^2+8)+3))`

`=lim_(x->-1)(x^2+8-9)/((x+1)(sqrt(x^2+8)+3))`

`=lim_(x->-1)(x^2-1)/((x+1)(sqrt(x^2+8)+3))`

`=lim_(x->-1)((x+1)(x-1))/((x+1)(sqrt(x^2+8)+3))`

`=lim_(x->-1)(x-1)/(sqrt(x^2+8)+3)`

`=(-1-1)/(sqrt((-1)^2+8)+3)`

`=-1/3`