Limit as X goes to negative Infinity of 1 -, X to the 4th over X ^2 + 5, X all to the 5th. If we thought about looking at the function and thinking about what the graph is going to do, we know that we're going to have vertical asymptotes when the denominator equals 0. So if I factor out an X, I'd get X * X + 5 equaling 0. So the vertical asymptotes are going to occur when X is 0 and X is -5. Looking for a horizontal asymptote or an oblique, the degree on top is bigger, so we know we're going to have to have an oblique. Actually it's going to be a parabola because the degree on top is too bigger. We thought about doing some long division here. Negative X to the 4th plus zero X ^3 + 0 X squared plus zero X + 1 divided by that X ^2 + 5 X. So we would get a negative X ^2 when we multiply, so negative X ^2 times that X ^2 -, 15 X would give us negative X to the 4th and positive 15X cubed. If we subtract, we bring down the next term. So now we're going to get a -15 X so -15 X cubed plus 225 X squared. When we subtract, we get -225 X squared and bring down the next term and leave myself quite enough room up here. So now we're going to have subtract 225 and so -225 X squared plus something big. 225 * 15225 * 15 is 3375 and that's positive. Oops, positive X. If we bring down or subtract and we bring down the one, that whole portion's our remainder because the degree is smaller than the X ^2. So our other asymptote in this case would be Y equal negative X ^2 -, 15 X -225 all to the fifth power. So if I have a negative and I take it to the fifth power, a negative coefficient to the fifth power is going to give me out a negative for a really, really big positive number. If we thought about X intercepts, our X intercepts occur when our numerator is equal to 0. So we'd have 1 -, X ^2 if we factored times 1 + X ^2. That 1 - X ^2 factors into one minus X 1 + X 1 + X ^2 doesn't factor because it's a sum of squares. So X is going to be 10 and -1 zero for our intercepts. So from there we have Y intercept isn't going to exist because when I put zero in for X, it all goes away. I'm going to erase my division over here so I have some room to do the graph. So when we look at this problem, we're going to type in our vertical asymptotes. We're going to type in our intercepts and our oblique asymptote. So vertical asymptote zero and also at -5. Once again, not to scale my any stretch of the imagination, an oblique asymptote is going to be a parabola X intercept, Y intercept. I'm sorry. It's going to be at -225 to the fifth power somewhere way down here. And that parabola is actually going to be to the fifth power. So a parabola to the fifth power is going to be to the 10th power, and the negative in front and negative times a -5 times is actually going to tell us that this 10th degree is going to go down. Hence really, really, really not to scale something like that. And we had X intercepts at 1:00 and also at -1 So when we're close, oh, this should have been dotted. So when we were close to Infinity, we've got to be close to this asymptote. So from here we're going to go to the X intercept. That X intercept came from this term, which has an odd multiplicity. So if the YS are negative on one side, they're positive on the other. That was all to the fifth power. So that's still true. This asymptote came from this term, which is on multiplicity to the fifth power, which is still an odd multiplicity. So if the YS on one side are negative, the YS on the other are positive. Remember, sorry, if the YS are positive on one side, they're negative on the other. Remember, we can cross an other asymptote or an oblique. We can't cross a vertical. This X intercept came from this term here. That was X + 1 one plus X. It's an odd multiplicity to an odd power. So if the Y's on one side are negative, the Y's on the other are positive. This vertical asymptote came from here. It's an odd multiplicity to an odd power of five. So if the Y's on one side are positive, the Y's on the other are negative, and then I've got to get close to that oblique asymptote as I go out to negative Infinity. So the limit as X goes to negative Infinity looking out here to the left is going to be negative Infinity for my Y values. I could have given other questions like what did what happens when limit as X goes to Infinity or zero from the left, zero from the right -5 from the left -5 from the right, one from the left, one from the right -1 from the left, negative +1 from the left and right. And then you could also be given any point and to get the point, you would just literally stick it in. So once we have the graph, it's very powerful because there are lots of different possibility, possible questions that could be asked from it.