Question

need a little help over here.​

Answer 1

Step-by-step explanation:

We have,

$\lim_{x - > 0} ( \frac{ \sqrt{1 + x + {x}^{2} } - 1}{x} )$

Rationalizing the denominator,

$= \lim_{x - > 0}( \frac{( \sqrt{1 + x + {x}^{2} } - 1)( \sqrt{1 + x + {x}^{2} } + 1) }{x( \sqrt{1 + x + {x}^{2} } + 1) } )$

$= \lim_{x - > 0}( \frac{1 + x + {x}^{2} - 1}{x( \sqrt{1 + x + {x}^{2} } + 1) } )$

$= \lim_{x - > 0} (\frac{x(x + 1)}{x (\sqrt{1 + x + {x}^{2} } + 1) } )$

$= \lim_{x - > 0}( \frac{x + 1}{ \sqrt{1 + x + {x}^{2} } + 1} )$

$= \frac{0 + 1}{ \sqrt{1 + 0 + 0} + 1}$

$= \frac{1}{2}$