Question

Limit x tends to infinity
Root x^2+x+1 - root x^2+x-1

Answer 1

Step-by-step explanation:

We have,

$\lim_{x \rarr \infty } \sqrt{ {x}^{2} + x + 1 } - \sqrt{ {x}^{2} + x - 1 }$

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

$= \lim_{x \rarr \infty } \frac{ {x}^{2} + x + 1 - {x}^{2} - x + 1 }{ \sqrt{ {x}^{2} + x + 1 } + \sqrt{ {x}^{2} + x - 1 } }$

$= \lim_{x \rarr \infty } \frac{1}{ \sqrt{ {x}^{2} + x + 1 } + \sqrt{ {x}^{2} + x - 1 } }$

$= 0$