Find the following limit if it exists, or explain why it does not exist:
lim as x approaches -infinity of square root of (9x^6-x^2) divided by x^3+5
lim x->-∞ √(9x^6-x^2)/(x^3+5) = ∞/∞ so we use derivatives a few times to get
As x gets huge, f(x) just looks like
√9x^6 / x^3 = -3
to get formal,
-lim √[(9x^6-x^2)/(x^6+10x^3+25)]
= √ lim [(9x^6-x^2)/(x^6+10x^3+25)]
Now use l'Hospital's Rule 3 times to evaluate the limit inside the radical
To find the limit, we can analyze the behavior of the function as x approaches negative infinity. We will break it down into two parts: the numerator and the denominator.
First, let's look at the numerator: √(9x^6 - x^2).
As x approaches negative infinity, the term x^6 dominates the expression since it has a higher power than x^2. Additionally, since x is negative, x^6 is always positive. Therefore, as x becomes more and more negative, the value of the term 9x^6 gets larger and larger.
Next, let's analyze the denominator: x^3 + 5.
As x approaches negative infinity, x^3 dominates the expression since it has a higher power than 5. Additionally, as x becomes more and more negative, x^3 becomes more and more negative, and the term 5 becomes negligible in comparison.
Now, let's consider the whole expression: (√(9x^6 - x^2)) / (x^3 + 5).
As x approaches negative infinity, the numerator grows without bound while the denominator approaches negative infinity. However, since both the numerator and denominator approach negative infinity at different rates, the limit of the expression does not exist.
In other words, the limit as x approaches negative infinity of (√(9x^6 - x^2)) / (x^3 + 5) does not exist because the numerator and denominator approach negative infinity, but at different speeds.