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.**