Find the limit or show it does not exist as x approaches inf

Solution

Rationalizing, the given function becomes

[ x^2 - ( x^2 + 2x ) ] / sqrt [ x - sqrt ( x^2 + 2x ) ]

= - 2x / [ x - sqrt ( x^2 + 2x ) ]......divide N and D by x = rt ( x^2 )

= - 2 / [ 1 - sqrt ( 1 + 2/x ) ]

As x --> inf., the limit of this function becomes

L = - 2 / [ 1 - ( 1 + 0 ) ]

= - 2 / 0, i.e., -infinity, which does not exist.

Hence, the req\'d limit does not exist.

[ 2 ] After rationalization, this function becomes

= x / [ sqrt( x^2 + x ) + x ].....divide N and D by x = rt ( x^2 )

= 1 / [ sqrt( 1 + 1/x ) + 1 ].

As x --> inf., its limit becomes

L = 1 / [ sqrt( 1 + 0 ) + 1 ]

= 1 / ( 1 + 1 )

= 1 / 2.