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Find the limit or show that it does not exist.

$ \displaystyle \lim_{x \to -\infty}\frac{\sqrt{1 + 4x^6}}{2 - x^3} $

2

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

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this is problem number twenty four of this through a calculus Eighth edition, Section two point six. Find the limit or showed that it does not exist. The limit is experts is negative. Infinity of the square root of the quantity one pleasant for X six two Ready brother Quantity to minus x Cute. Now our virtual involved wanting to find terms of the form one over X to the O. R. Where are you rational value that is created in zero? Because we know this limit is equal to zero as experts is either infinity or negative Infinity. So one approach her one way. We're gonna try to certify this is two first try to simplify the top a bit. The numerator and we're going to first is we're going to a factor out. Excellent. Sixth here. So are we. Fact, except six, which is the same as one over excellence. Six as the first term now plus for on divided by still two minutes to execute. So over here we have and we can write this over here. Scientist as we have the square would have thanks to the sixth. Now this is the same as the square root of X squared cubed and if we work on the square root of X squared is equal to the absolute value picks. Now here's where we have to use this. We need to understand this in this way. In order for us to account for the terms of X that will most definitely be negative. So we're dealing with the negative domain. Two we haven't now is absolutely of X cubed. And we also recalled that for X is less than zero. You have. So Volume X is equal to Negative IX exit less ten zero and that's her. Don't mean so. If this is true, then we replace upset Olympics with Negative IX, which cubed or means negative. So we now have reduced this creative excuse to this expert work too negative x cubed And that was a very important step. We cannot return to Parliament as experts thing to infinity, having a reduced it to negative X Cube. Tim, it's this quantity one of Rex with six plus four and the denominator we have to win is X cubed and we proceeded to divided by X cubed to the numerator and the denominator. Which leaves us with negative one of her exes. Six. Power this for fine. Two over. Execute. Find this one. And now we can use their definition. Any terms of the form one of her exes R. Where are the Russians? By agreement with zero is going to approach zero as experts native community. So through our properties of limits, we can treat each of these terms a separate limits on through properties of limits, having to do with partakes questions, some and differences. And so one of Rex and six will vanish to ever execute is going be negligible as experts Negative infinity. So our limit will reduce to negative the square root of four too crowded by negative one, which is thanks to develop a native form or too. And that is our final answer for the summit.

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