Find the following limit if it exists, or explain why it does not exist:

Question

lim as x approaches -infinity of square root of (9x^6-x^2) divided by x^3+5

Answer 1

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