We want to look at limit as X approaches zero from the right hand side of X ^2 / 9 - 7 / X So the first thing I'm going to do is I'm going to think about putting it as a fraction getting common denominator. So X ^3 - 63 / 9 X. If I graph just the function X ^3 - 63 / 9 X, then I ought to be able to look at the graph and come up with any of the limits. So first step, the vertical asymptote is where the denominator is not equal to 0, so X = 0 for a vertical asymptote. The next step X intercept, that's where the numerator equals 0. So add 63 and cube root it. So the cube root of 63, zero. There is no Y intercept because if I stick zero in for XI get 0 - 63 / 0, the oblique asymptote, or in this case actually it's an other asymptote. If I divide X ^3 - 63 by 9, XI get one 9th X ^2 - 63 / 9 X, Well, the 63 / 9 X when we're way at an Infinity or negative Infinity is going to go to 0. So in this case, we're going to have an other asymptote of Y equaling one 9th X ^2. If I put all those points on the graph, that's going to be enough to tell me how this graph should look when I'm way out of Infinity. I've got to be close to the asymptote at this point. Cube root of 63, zero, that's going to have an odd multiplicity. So it's got to go through and I've got to get close to this other asymptote. Remember, none of my graphs are to scale, not to scale. So if we look at this asymptote, its degree it came from the bottom here is odd. So if the Y values on one side are negative, the Y values on the other side have to be positive. So now when I look at this problem and I want to think about what happens to X as we go to zero from the right hand side. Well, if we're coming in to X approaching zero from the right hand side, what's happening to the YS? The YS are going out to negative Infinity. Looking at the remaining parts of this problem, if we want to know what happens to 0 coming in from the left side. So we are really asking what's happening to our Y values as we get closer and closer and closer to 0 on the left, we're going to have Infinity. What happens to X as we get closer and closer to the cube root of 63? And because it doesn't tell us left or right, the value has to be going to the same value for the Y. So if I'm looking going closer and closer and closer from the left and closer and closer and closer from the right, the Y value is getting to zero. We knew that because it was our X intercept. Now what happens when X goes to 63 somewhere way out here? Well, we would expect a positive value because 63 is getting big and it's going to go out to the asymptote and literally all we do is stick 63 in. So 63 ^2 / 9 - 7 / 63 and doing some plug and chug we get 3968 / 9. When we reduce the 763 really reduces to one ninth. Oops, that looked pretty awful. That 763 is really one ninth, so that's how we get ninths. 63 ^2 is going to be 3969 over 9 and so when I take one away from it I get the 3968.