Determine

Question

Determine $\lim_{x\rightarrow\infty}f\left(x\right)$ and $\lim_{x\rightarrow-\infty}f\left(x\right)$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = 4 x (3 x - \sqrt{9 x^{2} + 1})$

Accepted Answer

Welcome back everyone. In this problem, we want to solve the following limits and find the function of the horizontal asymptotes. If any for the function F FX equals two X multiplied by five X minus the square root of 25 X squared plus four. The limit of F FX as X approaches infinity as X approaches negative infinity. For our answer choices A says the limit as X approaches infinity is 4/5 as it approaches negative infinity is negative 4/5 and the horizontal asymptotes are Y equals 4/5 and Y equals negative 4/5. B says that the limit as X approaches infinity as it approaches negative infinity and the horizontal asym Y equals zero C says they are infinity. Well says the limit of both are infinity and there is no horizontal Asymptote. And D says the limit as X approaches infinity is negative for 1/5 as it approaches negative infinity is infinity. And the horizontal Asymptote is Y equals negative for 1/5. Now, how can we solve for the limits and the function of the horizontal asymptotes? Well, one thing first before we start, we can note that if X is positive. OK. Then the square root of X squared equals X. But if it's negative less than zero, then we take the square root of X squared as negative X. And now that's gonna come in a bit handy as we continue to work. OK. But before we do that, the first thing we need to do, we want to try to simplify the function F FX, we want to write it differently. So let's take our function F FX. OK. And first, let's see if we can multiply our expression by the conjugate of the numerator or the conjugate of our term. Uh That's in the bracket. OK. So four F of X being equal to two X multiplied by five X minus the square root of 25 X squared plus four. Let's write this as a quotient divided by one and then multiply it by five X plus the square root of 25 X squared plus four. OK. Divided by five X plus the square root of 25 X squared plus four. Now, when we go ahead and multiply by the conjugate here, OK, then we're gonna get two X multiplied by 25 X squared minus 25 X squared plus four. And that's all going to be divided by five X plus the square root of 25 X squared plus four. And we can simplify that even further because 25 X squared minus 25 X squared equals zero and two X multiplied by negative four. OK will be equal to negative eight X. So we can write this as negative eight X divided by five X plus the square root of 25 X squared plus four. Now we can take it a step further by dividing each term by X depending on the values of X. OK. So starting with the limit of FX as X approaches infinity get, then we can say or we can rewrite it here. We can divide through by X to make our expression become X approaches infinity of the negative value of eight X divided by X all divided by five X divided by X plus the square root of 25 X squared, divided by X squared plus four divided by X squared, eight X divided by X or X terms with cancer. The same with five X divided by X and 25 X squared divided by X squared. OK. And now that leaves us with a negative value of it divided by five plus the square root of 25 plus four divided by X squared. And again, we're finding the limit of that. So I should actually carry this term down here. OK. And let's just write that properly here. So it's equal to the limit of that as X approaches infinity. Now, when we, when we solve here, OK. And again, we've divided our radical expression by positive X, OK. Because X is greater than zero then no OK, as X approaches infinity four divided by X squared approach is zero. So really then this is going to be equal to negative eight divided by five plus the square root of 25 square root of 25 is five. So that's negative A divided by five plus five which is 10, which equals negative 4/5. So the limit of F FX as X approaches infinity is negative for fifth. So let's write that here. OK. Now what about the horizontal Asymptote? Well, that also means that since our limit as X approaches zero is negative for fifths, that means the function will also have a horizontal Asymptote. OK? At Y equals negative 4/5. OK. So that is going to be our solution to the first part of our problem. Now let's put the second part in blue. What about when we're trying to find the limit as X approaches negative infinity of F FX? Well, no F of X? OK. If we do the same, if we divide through by X, OK, where we divide our radical expression by negative X because as X approaches negative infinity, that means X is going to be negative, then our X is going to be less than zero. Then we can say then that the limit as X approaches negative infinity of the negative value of eight X divided by X. OK. All divided by five X divided by X plus. OK. Oh Sorry minus. OK. Because remember we're dividing our radical expression by negative X. So that's minus 25 X squared divided by X squared plus four divided by X squared. OK. Now again, it's gonna be similar to our previous one. So that's the limit as X approaches negative infinity of eight divided by five minus the square root of 25 plus four divided by X squared. Now again, in this case, we know that as X approaches negative infinity, the denominator will approach zero from the left. OK. So now in that case, OK. So let's just make that statement here as X approaches zero, uh sorry, as X approaches negative infinity, OK, then five minus the square root of 25 plus four divided by X squared approaches zero from the lift. OK? And since it approaches zero from the left, we can rewrite our limit as the limit as X approaches negative infinity of the negative value of a divided by zero approaching from the left. And that is gonna be equal to infinity. OK. So what this tells us then OK is that the limit as X approaches negative infinity of F FX will be equal to infinity. And since this limit is not a, a finite value, OK. That means the only horizontal, aim to it, the only horizontal I seem to is also then going to be Y equals negative for fifths. Therefore, this tells us then that the limit as X approaches infinity of FX is negative 4/5 as it approaches negative infinity, it equals infinity. And the horizontal Asymptote is Y equals negative 4/5. When we look back at our answer choices, that means D is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

Determine $\(\lim\)_{x\(\rightarrow\]\infty\)}f\(\left\)(x\(\right\))$ and $\(\lim\)_{x\(\rightarrow\)-\(\infty\)}f\(\left\)(x\(\right\))$ for the following functions. Then give the horizontal asymptotes of $f$ (if any).

$f (x) = 4 x (3 x - \sqrt{9 x^{2} + 1})$

Key Concepts

Limits at Infinity

Horizontal Asymptotes

Dominant Terms in Functions

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