Analyze the following limits and find the vertical asymptotes ...

Question

Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x⁴ − 49x²).

lim x→0 f(x)

Accepted Answer

Hi everyone, let's take a look at this practice problem dealing with limits. This problem says consider the function F of X is equal to the quantity of X + 9 in quantity, divided by the quantity of X-rays to the 4th minus 81 X2. Determine the limit as x approaches 0 of F of X and identify the vertical asymptotes of F. We're given 4 possible choices as our answers. For choice A, we have the limit as X 0 of F of X is equal to infinity, and the vertical asymptoes are at X equal to 0 and X equal to 9. For choice B, we have the limit as x approaches 0 of F of X is equal to minus infinity, and the vertical asymptote is at X is equal to -9. For choice C, we have the limit as X approaches 0 of F of X is equal to minus infinity, and the vertical asymptotes are located at X equal to 0 and X equals to 9. And for choice D, we have the limit as X approaches 0 of F of X is equal to -9, and the vertical asymptotes are located at X equals to 9. Now, this question asks us to determine two things. First, we need to determine a limit, and then we need to determine the vertical asymptotes. Let's focus on the limit first. So we're going to be looking at the limit as X approaches 0 of F of X, and we actually have our function F of X defined in the problem. So we're going to be looking at the limit. As X approaches 0 of the quantity of X plus 9 in quantity, divided by the quantity of X-rays to the 4th minus 81 X2. Now, before I actually take this limit, I want to actually simplify the denominator. So, if I look at the denominator, I can factor out an X2 out of both of those terms. This means that our limit is going to be equal to the limit. As X approaches 0, of the quantity of X plus 9 in quantity, divided by the quantity of X2, multiplied by The quantity of X2 minus 81. Now, if I look at the second term in our denominator, the X2 minus A1, that is a difference of 2 squares. So I can factor that. So, this is going to be equal to the limit. As x approaches 0 of the quantity of X plus 9 in quantity, divided by the quantity of X2, multiplied by the quantity of X minus 9. In quantity, multiplied by the quantity of X plus 9. Now, the factor of X + 9 will cancel out with the factor of X + 9 in the numerator. And so this limit is going to be equal to the limit, as X approaches 0. Of 1 divided by the quantity of X2 multiplied by the quantity of X minus 9. So, now, when I try to evaluate this limit, I noticed that I have a non-zero value in the numerator, but it's going to be divided by 0. So that means our limit here is going to approach plus or minus infinity, since my um fraction here is going to get arbitrarily large as X approaches 0. Now, if I look at the denominator, I have a factor of X2, which is always going to be positive, and the other factor is X minus 9, which is always going to be negative, while X is in the vicinity of 0. So I have a positive value multiplied by a negative value in the denominator. So overall, my denominator is going to be negative. And since my numerator is always positive, this value for the limit is going to actually approach minus infinity. So, this limit evaluates 2 minus infinity. And so, that is part of the answer that we're looking for for this question, that the limit as X approaches 0 of F of X is equal to minus infinity. So now we need to actually identify the vertical asymptotes. Now, we call for a vertical asymptote to be present, we need our denominator to be equal to 0, while we have a non-zero numerator. Now, since we look at the expression, The fraction that we had right before we evaluate our limit, we have a 1 on the numerator, which is always going to be non-zero, and then we can just set our denominator equal to 0 and figure out what values of X correspond to the denominator equaling 0, and those can be the locations of our asymptotes. So, we're gonna be looking at the quantity of X2d, multiplied by the quantity of X minus 9 in quantity is equal to 0. So, this gives us two solutions. We have X2d is equal to 0, which means that X is equal to 0. And we have X minus 9 equal to 0, and when we solve for this by adding 9 to both sides, we'll get X is equal to 9. And so those are the two locations of our vertical asymptotes. And we already saw that X equal to 0 was a vertical asymptotes since the limit as X approaches 0 of F of X approached minus infinity. So, now that we have um all of our answers identified for this question, we can come over to our answer choices, and the correct answer choice corresponds to answer choice C. So I hope that this explanation has been useful, and I'll see you in the next video.

Analyze the following limits and find the vertical asymptotes of f(x) = (x + 7) / (x⁴ − 49x²).
lim x→0 f(x)

Key Concepts

Limits

Vertical Asymptotes

Factoring Polynomials

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