Horizontal and Vertical Asymptotes
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Question

Horizontal and Vertical Asymptotes

Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .

Accepted Answer

Welcome back, everyone. Consider the function F of X equals square root of 6 X2 + 19 divided by 3 x minus 4. Identify the horizontal and vertical asymptotes of the graph of F of X. So what we're going to do is simply begin with the vertical asymptote. And we have to recall that if we have a rational function, we simply want to make sure that our denominator is equal to 0, so we said 3 x minus 4 equals 0, but we also want to make sure that for this value of X, which satisfies 3 x minus 4 equals 0, our numerator is not equal to 0, right, because we don't want to have a hole in the function for. A vertical asymptote to exist. So now what we're going to do is solve this equation 3 X equals 4, and therefore X is equal to 4/3. Now, does this make the numerator equals 0? Well, it doesn't, because we have square root of X2 + 19. We have to understand that X2 is always greater than or equal to 0. And we're adding 19, so the minimum value under the radical is essentially 19, right? It is greater than 0, so there is no x value which can make the numerator equal to 0, and this means that we have evaluated the vertical asymptote. Now when it comes to the horizontal asymptotes, we want to evaluate. Two limits. One of them is the limit as X approaches infinity of F of X. So we're going to rewrite the F of X 6 X2 + 19. Squire root of that. Divided by 3 X minus 4. If we just attempt the direct substitution, we will notice that we're getting infinity divided by infinity. So if we have the rational function, we basically want to divide all of our terms by the X with the highest exponent in the denominator, and that's simply X, right? Before we do that, we're also going to factor out X2d under the radical, so we're going to get square root of. X 2 in 6 + 19 divided by X2. And now for our denominator we still have 3 x minus 4, and we can also state that this is equal to the limit as x approaches infinity of the absolute value of X we're taking square root of X2 multiplied by square root of 6 + 19 divided by X2. And now in the denominator so far nothing changes. 3 x minus 4. And from here, let's just divide all of our terms by X. So we're going to get limit as X approaches infinity. Of the absolute value of x square root of 6 + 19 divided by x 2. If we're dividing both sides by X, we get 3 X divided by X, which is 3. We are dividing -4 by X, which gives us -4 divided by X, and we also have to include The division by X and the numerators, we can just add it under the absolute value of X. Well done. Now, from here, we have to understand that X approaches positive infinity, so the absolute value of X is just X itself. Right? Because if X is greater than or equals to 0, then the absolute value of X is just X. And now x divided by X gives us a value of 1. And now if x approaches infinity, our fractional terms are going to approach 0 in each case because a number divided by infinity is simply a limit of 0. And this gives us our first horizontal asymptote which is square root of 6 divided by 3. Now, the second limit that we want to evaluate for the horizontal asymptotes is limit as X approaches negative infinity. So we can just begin with the absolute value of X, right? So now what happens is that if X approaches negative infinity, we have to add a negative sign in front of X if we decide to drop the absolute value, right? So we're going to get negative. X multiplied by square root of 6 + 19 divided by X squad in the numerator, and for the denominator, we have 3 X minus 4. Now, once again, we're going to divide both sides by X and in this case, if we divide by X. Our numerator becomes negative X divided by X, which is -1, so we just have that additional negative sign, and our denominator is the same as before. It's just 3. Minus 4 divided by X. And in this case, once again our fractional terms, they approach 0 because we're dividing a number by an infinitely large number. It doesn't really matter if it's negative infinity or positive infinity. The limit is still going to be equal to 0. And the main difference here is that we get that negative sign. So the answer would be negative square root of 6 divided by 3, which is our second horizontal asymptote. So we can conclude that the vertical asymptote is X equals 4/3, and the horizontal asymptotes are Y equals positive or negative square root of 6 divided by 3. Those would be our answers to this problem. Thank you for watching.

Horizontal and Vertical Asymptotes
Assume that constants a and b are positive. Find equations for all horizontal and vertical asymptotes for the graph of y = (√ax² + 4) / (x―b) .

Key Concepts

Horizontal Asymptotes

Vertical Asymptotes

Rational Functions

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