Problem 74 Hard Difficulty

Let $ f(x) = (x^3 + 1)/x $. Show that
$$ \displaystyle \lim_{x\to \pm \infty} [f(x) - x^2] = 0 $$
This shows that the graph of $ f $ approaches the graph of $ y = x^2 $, and we say that the curve $ y = x^2 $. Use this fact to help sketch the graph of $ f $.

Answer

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Video Transcript

in this problem, we want to show that given f of X is equal to execute plus one over X that first, the limit as Ex purchase plus board minus infinity of f of X minus X squared is equal to zero. And what this will show us is that on either side of our graph is going to approach the same shape as X squared. And then we're gonna use this fact to help us graph our function f of X. But so first, let's just go ahead and find what f of x minus X squared iss. So let's do that first, since we're gonna need to use it for two limits. So we go ahead and plug in X cubed plus one all over X minus specs squared. We'd go ahead and get a common denominator here by multiplying this by ex that would give us execute. Then we'd have X cubed plus one minus x cubed all over X. The x cubes cancel here and then we're left with one over X. So if we come over here to this limit that we're trying to find so first, just the limit as X approaches infinity of f of X minus X squared, which ended up just being one over X. Well, we know that goes to zero and then likewise the limit as this approach is negative affinity would also go to zero as well. So we now have this which shows that when we go to graft this, it'll have the same in behavior as X squared. So I've gone ahead and made a graph of X squared really quickly. Just so when we are graphing this we will be able to just kind of follow it really closely. Now, the other things that might be of interest at least since we're just sketching it, is we'd want to say where our intercepts are going to be and any horizontal Ason Phillips we're going to have, um, well, vertical acid trips because we already know we're gonna have no horizontal because the end behavior follows closely to X squared. So to find our intercepts, So let's go ahead and do the ex intercepts. So remember, we set f of X equal to zero, so x intercept. So we have f of X is equal to zero, which means we have x cubed plus one over X equals zero. Then we would just set the numerator equal to zero. Subtract one. Then we would take the cube root on each side. They keep root of negative one is just gonna be negative one. So we have X is equal to negative one for one of our troops. Let's go ahead and plot that point. So we're right there. Next we should find our vertical asking too. So remember, for vertical awesome times we just set the denominator equal to zero. So vertical asked him to Well, that would just be X is equal to zero since that's one thing we have in our denominator. So let's go ahead and put that down. And now we need to decide if we're going to be on the right or the left of this function so we can go ahead and take the limit as X approaches zero from the right of execute plus one all over X. So, plugging this in here we would end up with zero fromthe right, cubed plus one all over zero from the right. And now something from the right is going to be positive if we keep it. So that's still going to be zero from the right, plus one all over. Still something zero from the right. So adding something coming from the right zero and one would just give one. We have one over his ear from the right, and so that would go to positive infinity. So we know there. Our function should start coming up like this here. And since this vertical ask himto is like one over X, we know that the limit on the other side, as we approach zero fromthe left, should be over here where it starts. So that means at least on the rate, it'll look something kind of like this. I was going to dip down, and then at some point, let me draw that a little bit better. So it was going to dip down and then come back up at some point and that it will start following X squared really closely and then on this other side, we're gonna come up from negative infinity, We're gonna have to cross our X intercept, and then we're going to get very close to X squared and then just fall this out, too. And so this is how we could go ahead and use that this function approach is essentially the same as X squared to go ahead and get a nice sketch of this crap.