Question

`lim_(x → 0) (sqrt(1 + x + x^2) − 1)/x` = ______.

Options

• `1/2`

• `−1/2`

• 0

• ∞

Answer 1

`lim_(x → 0) (sqrt(1 + x + x^2) − 1)/x` = `bbunderline(1/2)`.

Explanation:

By rationalisation of the numerator, given expression 

= `lim_(x → 0) (sqrt(1 + x + x^2) − 1)/x.(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(1 + x))/(x(sqrt(1 + x + x^2) + 1))`

= `lim_(x → 0) (1 + x )/(x(sqrt(1 + x + x^2) + 1)) = 1/2 z`