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) = (x⁴ − 1)/(x^2−1)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with limits. This problem says evaluate the limit as x approaches plus or minus infinity and identify any horizontal asymptotes for the function F of X is equal to the quantity of X 4 minus 25 in quantity divided by the quantity of X2 minus 25. We're given 4 possible choices as our answers. For 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 infinity, and there are no horizontal asymptotes. 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 FX is equal to minus infinity, and there are no horizontal asymptotes. 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 FX 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 1, the limit as X approaches minus infinity of F of X is equal to -1, and the horizontal asymptotes are Y is equal to 1, and Y is equal to -1. Now, this question asks us to do two things. First, to evaluate the limit as X approaches plus or minus infinity, and to identify any horizontal asymptotes. So, we need to determine the limits first in order to determine the asymptotes. So we're gonna start there. So we're gonna be looking at the limit. As x approaches. Plus or minus infinity of our function F of X, which is the quantity of X minus 25 in quantity, divided by. X2 minus 25. So, if we try to evaluate this limit by direct substitution, we get the indeterminate form of infinity divided by infinity. So, we need to use another method to determine this limit, and the method that we're going to use is to divide both the numerator and denominator by the highest power of X in the denominator. In this case, that's going to be X2d. So doing so, this is going to be equal to the limit. As x approaches, plus or minus infinity of the quantity of X fourth divided by X2 minus 25 divided by X2 in quantity, divided by the quantity of X2 divided by X2 minus 25 divided by X2. So, we can simplify this expression, so this is going to be equal to the limit. As x approaches plus or minus infinity. Of the quantity, we'll have X 4th divided by X2, which can be X2. Minus 25 divided by X2 in quantity. And that is going to be divided by the quantity of X2 divided by X2, which is 1 minus 25 divided by X2. Now notice that we have a fraction in both the numerator and denominator of the form of a constant 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 N is equal to 0 when N is greater than 0. So, we can use this information in order to evaluate the limit. And so we need to look at the limit for both the um positive infinity and negative infinity values. So we're gonna start off with the positive infinity. So, we're gonna look at the limit. As X approaches infinity. Of the quantity of X squad. Minus 25 divided by X2. In quantity, divided by the quantity of 1 minus 25 divided by X squared. So now when we take the limit, this is going to be equal to. I can substitute in infinity for X, and so in the numerator, the first term is going to be infinity squared, which is still just infinity. We'll have infinity minus the next term evaluates to zero. And then that quantity is going to be divided by the quantity of 1 minus 0. So simplifying this, this is just going to be equal to infinity. That's one of the limits that we needed to find. Now we need to do the same thing, or the X approaching minus infinity. So we'll have the limit. As x approaches minus infinity. Of the quantity of X2 minus 25 divided by X2 in quantity, divided by the quantity of 1 minus 25 divided by X2. And again, we can substitute N minus infinity for X, so this is going to be equal to. So, for the first time we'll have minus infinity squad, which gives me infinity squared, which again is just infinity. Then we'll have 0 in the numerator, and that quantity is going to be divided by the quantity of 1 minus 0. And so, again, this simplifies down to infinity. That is the other limit that we need to evaluate. Now, since both of these limits are evaluated to infinity, that means that we have no horizontal asymptotes. That's because they are not approaching any sort of finite value, which is what we need for a horizontal asymptote. So, we have no horizontal asymptotes in this case. So now that we have the limits, and we've identified the lack of horizontal asymptotes, we can now come up to our answer choices, and the correct answer choice actually corresponds to answer choice A. 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) = (x⁴ − 1)/(x^2−1)

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Rational Functions

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