Find the limit
lim
x→−∞
sqrt(4x^6 − x)/(x^3 + 6)
top ---> 2 |x^3|
because sqrt(x^6) is + but sqrt(x^3) is -
bottom ----- x^3 which is -
so 2 |x|^3 /-|x|^3 = -2
Thanx!!
To find the limit of the given expression as x approaches negative infinity, we can divide the numerator and denominator by x^3, which is the highest power of x in the expression:
lim(x→−∞) sqrt((4x^6 - x)/(x^3 + 6))
By dividing both numerator and denominator by x^3, the limit expression simplifies as follows:
lim(x→−∞) sqrt((4x^6/x^3 - x/x^3) / (x^3/x^3 + 6/x^3))
= lim(x→−∞) sqrt((4 - 1/x^2) / (1 + 6/x^3))
Now, as x approaches negative infinity, the terms (1/x^2) and (6/x^3) tend towards 0 since the denominator becomes infinitely large. Therefore, we can substitute 0 for these terms:
lim(x→−∞) sqrt((4 - 0) / (1 + 0))
= lim(x→−∞) sqrt(4/1)
= sqrt(4)
= 2
Therefore, the limit of the given expression as x approaches negative infinity is equal to 2.