When x gets very large, the -x inside the square root is insignificant in comparison to the 4x^6 so can effectively be ignored. The same thing about the 3 on the bottom in comparison to the x^3.
The effectively leaves sqrt(4x^6)/x^3 = 2
It is usually a good idea to check with a calculator.
x = 1000 ----> sqrt(4x^6 - x)/(x^3 + 3) = 1.999999994
This is not a proof of correctness but a good indication.
Find the limit
lim x→−∞ sqrt(4x^6−x)/(x^3+5)
2 answers
actually it should be -2
notice that x ---> negative infinity, so the x^3 is negative
notice that x ---> negative infinity, so the x^3 is negative
1 answer