Evaluate the following limit: `underset( x rarr 0) ( "lim")(sqrt(1+x+x^2)-1)/x`

Explanation

Given,

`(lim)_(x->0)(sqrt(1+x+x^2)−1)/x`

`=(lim)_(x->0)(sqrt(1+x+x^2)−1)/x xx (sqrt(1+x+x^2)+1)/(sqrt(1+x+x^2)+1)`

`=(lim)_(x->0)(1+x+x^2-1)/(x(sqrt(1+x+x^2)+1))`

`=(lim)_(x->0)(x+x^2)/(x(sqrt(1+x+x^2)+1))`

`=(lim)_(x->0)(x+1)/((sqrt(1+x+x^2)+1))`

`=1/2`