Evaluate `lim_(x->0)(sqrt(2+x)-sqrt(2))/x`

Explanation

When `xto0,` the expression `underset(xto0)lim(sqrt(2+x)-sqrt(2))/(x).` takes the form 0/0. Rationalizing the numerator, we have
`underset(xto0)lim(sqrt(2+x)-sqrt(2))/(x)=underset(xto0)lim((sqrt(2+x)-sqrt(2))(sqrt(2+x)+sqrt(2)))/(x(sqrt(2+x)+sqrt(2)))`
`=underset(xto0)lim(2+x-2)/(x(sqrt(2+x)+sqrt2))`
`=underset(xto0)lim(1)/(sqrt(2+x)+sqrt(2))=(1)/(2sqrt(2))`