Find the limit.?

Question

the limit as x---> infinity
of. root ((4x^2)-1)/ (x^2)

Answer 1

rationalize numerator
you get (4x^2-1)/(x^2sqrt4x^2-1)
=(4x^2-1)/sqrt4x^6-x^4

divide top and bottom by x^12
get 4/x^10-1/x^12/sqrt(4-1/x^8)
as approach inf
get 0-0/sqrt(4)=0
the answer is 0

Answer 2

this question is actually pretty simple. when the limit is approaching infinity there are three rules that you can use.

when the degree of x is the same on the top and the bottom the limit is the coefficients in front of x.

when the degree of the numerator is bigger than the degree of the denominator the limit is infinity

when the degree of the denominator is bigger than the degree of the denominator the limit is 0.

this question applies to the first rule. since both the x's are squared its just the coefficients so 4/1 = 4.

Answer 3

limit is 0.
factor x out of top and bottom:
√(4x² - 1) / x² =
x√(4 - 1/x²) / x² =
√(4 - 1/x²) / x, and as x→∞, you have
√(4 - 0) / ∞ = 2/∞ = 0

Answer 4

When x approaches infinity, subtracting 1 is insignificant, the (x^2)'s cancel each other. Meaning, as x approaches infinity, that equation approaches 4.