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 → -7 f(x)

Accepted Answer

Welcome back, everyone. Consider F of X equals x + 9 divided by X to the power of 4th minus 81 x2. Determine the limit as x approaches -9 of F of X and identify the vertical asymptote or asymptotes of the function. So, before we begin, we are going to look at the function F of X. It says X + 9. It is a rational function divided by X to the power of 4th minus 8181 X squad. Usually, if we can factor, we are going to factor any type of function before manipulating it, right? So let's apply this rule. We can carefully look at the denominator and we can see that we can perform some factorization. So let's begin. First of all, we can factor out X2, right? So if we take out X2, this leaves us with X2 minus 81 for the other factor. And now let's see that this is a difference of squares. We have X2 minus 9 squad, which can also be factored out using the formula for the difference of squares factorization. So we end up with three factors X2d. X minus 9 multiplied by X + 9. Now from here we have got a common factor which is X + 9, and if we cancel it out we're going to get a 1 in the numerator. In the denominator we're going to get Xquad multiplied by X minus 9, and we have to say that so far we want to make sure that X. is not equal to -9, right? Because this is exactly the point that we have crossed out by canceling out the factors, right? -9 is a whole because it is the common factor in the numerator and the denominator. So now we have our simplified form of the function, and we're going to look at the limit. So let's evaluate the limit. Limit as X approach is -9. And to identify if this limit exists, well, we want to make sure that the limit from the left and from the right gives us the same value, right? So let's substitute the function. We're going to get 1 divided by X squared multiplied by X minus 9. We can simply attempt the direct substitution. If the value is finite, we are done, right? And it might be finite because we have one in the numerator. What about the denominator? If we substitute -9 into the function. We're going to get -9 squared multiplied by -9 minus 9. And yes, indeed, this is going to be finite value, so we don't need to worry about -9 from the left or from the right. Let's simplify the result. We're going to get one in the numerator. In the denominator, we simply have 81. Or basically, we can say 9 square because negative 9. Squared is 9 squad where squaring a negative number multiplied by. -18, right? So we can factor out the negative sign and 18 is 2 multiplied by 9. So overall this is simply -1 divided by 2 multiplied by 9 cubed, and if we evaluate the result, this gives us -1 divided by 1,458. So we now have the answer to the limit. And finally we want to identify the vertical asymptotes. Let's recall that for a rational function p of X divided by Q X. To get vertical asymptotes, we want to make sure that our denominator is undefined for such x values. So we're setting it equal to 0, and for the same x values, our numerator should not be equal to 0, otherwise, Such an x value produces a hole, which is a removable discontinuity. Now, we already see that we have one hole which is X equals -9. This is because. This value makes the denominator and the numerator equal to 0. But now since we're left with 1 in the numerator, we can take care of the denominator. X squared multiplied by x minus 9 must be equal to 0 because it makes our denominator equal to 0. And because we have two factors, well, we can get two asymptotes. First of all, X2d can be equal to 0 or X minus 9 can be equal to 0. Solving these, we get either X is equal to 0 or X equals 9. And these two are the vertical asymptotes. Apparently they do not make the numerator equal to 0, so they are not holes. We can simply label them as the second part of the answer. Those are the vertical asymptotes. Thank you for watching.

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

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