Find all horizontal asymptotes for f(x) = (5x) / (x^2 + 3)^.5?

Question

The function is read as
"5x divided by [The square root of (x raised to the second + 3)]

Answer 1

f(x) = (5x)/(sqrt(x^2 + 3))

y = (5x)/(sqrt(x^2 + 3))
x = (5y)/(sqrt(y^2 + 3))
xsqrt(y^2 + 3) = 5y
sqrt(x^2(y^2 + 3)) = 5y
x^2(y^2 + 3) = 25y^2
x^2y^2 + 3 = 25y^2
x^2y^2 - 25y^2 = -3
y^2(x^2 - 25) = -3
y^2 = (-3)/(x^2 - 25)
y = sqrt((-3)/(x^2 - 25))
y = (sqrt(-3(x^2 - 25))))/(x^2 - 25)

y^2 - 25 = 0
y^2 = 25
y = -5 or 5

ANS : -5 or 5

The horizontal asymptotes are -5 and 5

Answer 2

The limit of f(x) when x tends to infinity is -5 for negative infinity and 5 for positive infinity. So the horizontal asymptotes are:
y=-5 and y = 5.