Find the limit
lim x→−∞ sqrt(4x^6−x)/(x^3+5)
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.
actually it should be -2
notice that x ---> negative infinity, so the x^3 is negative
To find the limit of the given expression, we can follow these steps:
Step 1: Determine the highest power of x in the numerator and the denominator.
- In this case, the highest power of x in the numerator is 6 (from 4x^6), and in the denominator is 3 (from x^3).
Step 2: Divide both the numerator and the denominator by the highest power of x.
- Dividing the numerator and denominator by x^3 will help simplify the expression:
lim x→−∞ sqrt((4x^6−x)/(x^3+5))
= lim x→−∞ sqrt((4x^6−x)/(x^3+5)) * (1/x^3) / (1/x^3)
= lim x→−∞ sqrt((4x^6/x^3−x/x^3)/(x^3/x^3+5/x^3))
= lim x→−∞ sqrt((4x^3−1)/(1+5/x^3))
= lim x→−∞ sqrt(4−1/x^3)/(1+5/x^3))
Step 3: Evaluate the limit as x approaches negative infinity.
- As x approaches negative infinity, the expression within the square root will approach 4 (the highest power of x in the numerator is dominant), and the expression within the denominator will approach 1 (as the denominator does not involve any negative powers of x).
- Therefore, the limit is:
lim x→−∞ sqrt(4)/(1) = sqrt(4) = 2
Therefore, the limit of the given expression as x approaches negative infinity is 2.