Investing rational functions horizontal asymptote

Write and simplify an expression for the time needed for the round trip as a function of the boat's speed. Express the team's time on the upstream leg as a function of the speed of the current. Write a function for the team's time on the downstream leg. Write and simplify an expression for the total time for the training run as a function of the current's speed.

Two pilots for the Flying Express parcel service receive packages simultaneously. The prevailing winds blow from east to west. Express Orville's flying time as a function of the windspeed. Write a function for Wilbur's flying time. Who reaches his destination first? So, let's multiply both the numerator and the denominator by one over x, or another way of thinking about it is we're dividing both the numerator and the denominator by x.

And if we're doing the same thing to the numerator and the denominator, if we're multiplying or dividing them by the same value, I should say, well then, I'm just really just multiplying it by one. So, I'm not changing its value. This will make it a little bit more interesting, and a little bit easier for us to think about what happens when x becomes very, very, very negative.

So, 7x-squared divided by x, or being multiplied by one over x, is going to be 7x. And then all of that over 15x divided by x, or 15x over x, is just going to be And then you have five over x. Five times one over x is equal to five over x. Minus five over x. Now, this is equivalent, for our purposes, to what we started with but it makes it a little bit easier to think about what happens when x gets very, very, very, very negative.

Well, when x gets very, very, very, very, very, very, very negative, this is going to become a very large negative number. You subtract two from it, it really won't matter much. You divide that by 15, well, that's not gonna matter much. And this is just going to become very, very, very small. You're taking five and you're dividing it by ever-larger negative numbers, or more and more negative numbers.

So, this right over here is gonna go to zero. This thing over here is gonna go towards infinity. Or, I should say, it's gonna go towards negative infinity. Seven times a negative trillion, seven times a negative googol, seven times a negative googolplex, we're getting more and more negative numbers, this is gonna get, this is going to approach negative infinity.

Doesn't matter that you're subtracting two from that. In fact, that'll get even more negative. And it doesn't matter if you then divide that by 15, you're still approaching negative infinity. If you had a arbitrarily negative number, you divide it by 15, you still have an arbitrarily negative number.

And, so, you could say that this is going to go to negative infinity. Now, another way that you could've thought about it. This is actually how I do think about it when I'm trying to, when I see these types of problems. I say, well which terms in the numerator and the denominator are going to dominate? And what do I mean by "dominate"? Well, as x gets very positive or x gets very negative, another way to think about it is the magnitude of x gets large, the absolute value of x gets large.

The higher degree terms are going to grow much faster than the lesser degree terms. And so, we could say that for large x, for large x, and when I say "large" I mean high absolute value. High absolute value. And if we're going to negative infinity, that's high absolute value. So, f of x is going to be approximately equal to the highest degree term on the top, which is 7x-squared, divided by the highest degree term on the bottom. So, as this becomes larger and larger and larger, this is going to matter a lot, lot less.

So, it's going to be approximately that. Which is equal to 7x over Well, even here, if you think about what happens when x becomes very, very negative here. Well, you're just gonna get larger, you're gonna get more and more and more negative values for f of x. So, once again, f of x itself is going to approach, is going to go to, negative infinity as x goes to negative infinity.

Let's do another one of these. So, here they're telling us to find the horizontal asymptote of q. A horizontal asymptote, you can think about it as what is the function approaching as x becomes, as x approaches infinity, or as x approaches negative infinity. And just as a couple of examples here. It's not necessarily the q of x that we're focused on. But you could imagine a function, let's say it has a horizontal asymptote at y is equal to two, so that's y is equal to two there.

Let me draw that line. So, let's say it has a horizontal asymptote like that. Well then the graph could look something like this. It could look, let me draw a couple of them that have horizontal asymptotes. So, maybe it's over here, it does some stuff, but as x gets really large, it starts approaching, the function starts approaching that y equals two without ever quite getting there.

And it could do that on this side as well. As x becomes more and more negative.

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Formally, horizontal asymptotes are defined using limits. If the degree of P x is equal to that of Q x , f x has a horizontal asymptote that is the ratio of the coefficients of the highest degree term of P x to that of Q x. If the degree of P x is greater than the degree of Q x , f x has no horizontal asymptote, though it may have a slant asymptote if the degree of P x is 1 greater than that of Q x. Examples Find any horizontal asymptotes for the following functions: 1.

The degree of P x is 2 and the degree of Q x is 3. This corresponds to the first case described above, where the degree of Q x is greater than that of P x. The degree of P x is 4 and the degree of Q x is 4. This corresponds to the second case described above, where the degrees of P x and Q x are equal.

Thus, f x has a horizontal asymptote at the ratio of the coefficients of the highest degree term of P x to Q x , or It is possible for a function to cross a horizontal asymptote. Let N be the degree of the numerator and D be the degree of the denominator.

Substitute in a large number for x and estimate y. Figure 8: No horizontal asymptote when the degree of the numerator is greater than the degree of the denominator. Slant Asmptotes Slant asymptotes occur when the degree of the numerator is exactly one more than the degree of the denominator. To find the equation of the slant asymptote, divide the fraction and ignore the remainder. Figure 9: Slant Asymptote when the degree of the numerator is 1 more than the degree of the denominator.

Notice that a graph of a rational function will never cross a vertical asymptote, but the graph may cross a horizontal or slant asymptote. Also, the graph of a rational function may have several vertical asymptotes, but the graph will have at most one horizontal or slant asymptote. In general, if the degree of the numerator is larger than the degree of the denominator, the end behavior of the graph will be the same as the end behavior of the quotient of the rational fraction.

Figure Quadratic Asymptote when the degree of the numerator is 2 more than the degree of the denominator. Horizontal Asymptotes of Rational Functions The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator. Slant Asymptotes of Rational Functions The slant asymptote occurs when the degree of the numerator is 1 more than the degree of the denominator.

The slant asymptote is found by dividing the rational function and ignoring the remainder. However, there is a slant asymptote because N is 1 more than D. Find the slant asymptote by dividing the rational function and ignoring the asymptote. Find the horizontal asymptote and interpret it in context of the problem.

In this case the cost will approach Solution Conveniently, this is already factored. Notice that there are no common factors between the numerator and denominator, so there are no removable discontinuities. Find the vertical asymptotes by setting the denominator equal to zero and solving for x. To find the horizontal asymptotes, check the degrees of the numerator and denominator.

Think of the result of multiplying the factors together.

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