Show that the functions f(x) = sqrt [ x^2 + 5) and g(x) = (2*sqrt x – 1)^2 grow at the same rate as x = inf.

We have to prove that the functions f(x) = sqrt (x^2 + 5)
and g(x) = (2*sqrt x – 1)^2 grow at the same rate as x
-->inf.

Two functions grow at the same rate if [lim
x--> inf ( f(x) )] / [lim x--> inf ( g(x) )] = constant not equal to
0.

Substituting the functions we have
here:

[lim x--> inf (sqrt (x^2 + 5))] / [lim
x--> inf ((2*sqrt x – 1)^2 )]

=> lim x
--> inf [(sqrt (x^2 + 5)/(2*sqrt x – 1)^2 ]

divide
the numerator and denominator by x

=> lim x
--> inf [[(sqrt (x^2 + 5)/x] / [(2*sqrt x – 1)^2/x]
]

=> lim x --> inf [[(sqrt (x^2/x^2 + 5/x^2)]
/ [(2*sqrt x / sqrt x – 1/ sqrt x)^2] ]

=> lim x
--> inf [[(sqrt (1 + 5/x^2)] / [(2 – 1/ sqrt x)^2]
]

As x --> inf , (1/x) -->
0

=> [(sqrt (1 + 0)] / [(2 –
0)^2]

=> 1 /
4

As 1/ 4 is a constant and not equal to 0,
the two functions grow at the same rate as x-->
inf.