Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any ...

Question

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x) = (x² − 9)/(x(x−3))

Accepted Answer

Hi everyone, let's take a look at this practice of problem dealing with limits. This problem says evaluate the limit as x approaches plus or minus infinity and identify any horizontal asymptotes where the function F of X is equal to the quantity of X2 minus 36 in quantity divided by the quantity of X multiplied by the quantity of X minus 6. We're given 4 possible choices as our answers. Choice A, we have the limit as X approaches infinity of F of X is equal to the limit as X approaches minus affinity of F of X is equal to 0. The horizontal asymptote is Y is equal to 0. For choice B, we have the limit as X approaches infinity of F of X is equal to the limit as x approaches minus infinity of F of X is equal to -1. The horizontal asymptote is Y is equal to -1. For choice C, we have the limit as x approaches infinity of F of X is equal to the limit as x approaches minus infinity of F of X is equal to 1, and the horizontal asymptote is Y is equal to 1. And for choice D, we have the limit as X approaches infinity of F of X is equal to infinity. The limit as X approaches minus infinity of F of X is equal to minus infinity, and there are no horizontal asymptotes. There are two parts to this question. The first part asks us to evaluate the limit as X approaches, um, plus or minus infinity or function, and then for the second part, we need to identify any horizontal asymptotes. So, we need to evaluate the limits first, in order to even find our horizontal ascetotes. And so, we have our function F of X here. However, we'll need to actually simplify this a little bit. So, we'll have F of X. is equal to the quantity of X2 minus 36 in quantity, divided by the quantity of X, multiplying in the quantity of X minus 6. And what we want to do is distribute that X in the denominator. So this is going to be equal to the quantity of X2 minus 36, nothing changed in the numerator in quantity, divided by the quantity of X2 minus 6 X. That's why I get on the denominator once I distribute the X. And so now we need to take the limit as X approaches plus or minus infinity. And if we just use direct substitution, plugging in either infinity or minus infinity and for X, we'll get a uh indeterminate form of infinity divided by infinity. So, in order to get rid of that indeterminate form, we're gonna need to divide both the numerator and denominator by the highest power of X. In this case, that is going to be X2d. So we're gonna have the limit. As X approaches, plus or minus infinity. Of the quantity of X2, divided by X2. Minus 36 divided by X2 in quantity, that's to be divided by the quantity of X2, divided by X2 minus 6 X divided by X2. So we can simplify this, so this is going to be equal to the limit. As x approaches, plus or minus infinity. Of the quantity, we'll have X2 divided by X2, which is 1. Minus 36 divided by X2. In quantity, divided by the quantity, again, we'll have X2 divided by X2, which is just 1, we'll have minus 6 X divided by X2, which is just 6 divided by X. Now that we have this in this form, notice that we have this fraction that has a value divided by some power of X. And so recall that the limit. As X approaches. Plus or minus infinity. Of the quantity Of a Divided by X to the N is going to be equal to 0, as long as N is greater than 0. So, we can use this um expression here to evaluate the limit. And so we're gonna start off by looking at the limit as X goes to the plus infinity. We'll have the limit. As x approaches infinity. Of the quantity of 1 minus 36 divided by X 2 in quantity, divided by 1 minus 6 divided by X. And so we can use the relationship that we just um looked at dealing with the limits, as X approaches plus or minus infinity of the form of a divided by X to the end, and that's exactly the form that those last two fractions in both the numerator and denominator have. So using that limit to evaluate this limit, this is going to be equal to 1 minus 0 in the numerator, and that quantity is going to be divided by the quantity of 1 minus 0 in the denominator. And just simplifying this expression, this is going to be equal to one. That's one of the limits that we needed to find. We're going to do the same thing with X approaching minus infinity. We will have the limit. As X Approaches minus infinity. Of the quantity of 1 minus 36. Divided by X 2 in quantity, divided by the quantity of 1 minus 6 divided by X. Again, apply the limit in both of these cases, those fractions will then also go to 0 when applied the limit. So this is going to be equal to the quantity of 1 minus 0 in quantity divided by 1 minus 0. And again, we can simplify this, and this is going to be equal to 1. So that is the other limit that I needed to evaluate. And finally, we need to determine the horizontal asymptotes. And in this case, since both of our limits are approaching 1, as we go to minus infinity or whether we go to plus infinity, that means that we're going to have a horizontal asymptote at Y is equal to 1. So, here we just have a single horizontal absentote with the function of Y is equal to 1. So, now that I have both the limits and my horizontal asymptote, I can look at the answer choices that I was given, and the correct answer choice actually corresponds to answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.
f(x) = (x² − 9)/(x(x−3))

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Rational Functions

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