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.
- print Print
- list Cite
Expert Answers
calendarEducator since 2010
write12,544 answers
starTop subjects are Math, Science, and Business
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.
Related Questions
- `G(y) = ln(((2y+1)^5)/(sqrt(y^2 + 1)))` Differentiate the function.
- 1 Educator Answer
- Given f(x) and g(x), please find (fog)(X) and (gof)(x) f(x) = 2x g(x) = x+3
- 1 Educator Answer
- Let f(x) = mx^2 + 2x + 5 and g(x) = 2x^2 - nx - 2. The functions are combined to form the new...
- 1 Educator Answer
- For the functions f(x) = x^2 and g(x) = (x-7), are f (g(x)) = g (f(x)). Also, what is g (f (2))?
- 2 Educator Answers
- If f(x)=x+4 and h(x)=4x-1, find a function g such that g(f(x)) = h(x).
- 2 Educator Answers
Unlock This Answer Now
Start your 48-hour free trial to unlock this answer and thousands more. Enjoy eNotes ad-free and cancel anytime.