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A function $ f $ is a ratio of quadratic functions and has a vertical asymptote $ x = 4 $ and just one $ x $-intercept, $ x = 1 $. It is known that $ f $ has a removable discontinuity at $ x = -1 $ and $ \displaystyle \lim_{x \to -1} f(x) = 2 $. Evaluate (a) $ f (0) $ (b) $ \displaystyle \lim_{x \to \infty} f(x) $

$\lim _{x \rightarrow \infty} f(x)=5$

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

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this problem. Number fifty nine of this tower Calculus A tradition section two point six Function is a ratio quadratic functions and, as a vertical asked him to x equals four and just one x intercept X equal to one. It is known that have has a remove with this. Continuity at X equals negative one and the limit is experts think it one of us is equal to two in early right party zero The party The Ltd's experts infinity of aftereffects So we have some description for what this function should look like. We'LL begin with mother clear hints when we discuss removable continuity Ese Sorry, removable discontinuities Are we talking about terms that cancel out in the numerator and the denominator? And if this removable discontinuity occurs at X equals negative one, this term should be X plus one. So we almost know this for sure. If we have a function similar to this the X plus one over the phone. If when factored out would be able to be cancelled, that would be what a removable discontinuity would it be like such that initially the function is there is on to find any other one but it is removed and therefore becomes a whole rather than an infinite discontinuity regarding the vertical aspirin, too. This has to do wait value that is undefined for the function but is not a removable. This continuity. And if we want to burn a cross in tow to occur at X equals two four, we need a term equal or exactly X minus war here and denominator. Therefore, one X equals four denominators equals zero and dysfunction undefined and therefore it is the critical asked him to because it is not removed. There's no removable termine the numerator. The next part we're going to satisfying is the X intercept. The ex interceptor and X intercept occurs a win. The function equals zero. So whenever we plug in about you in the function is equal to zero are that value is known as an exit in Jersey and in this case, the eccentricity sequel to Ex Eagle one. Therefore, when X equals one, this function should equal zero. One way to assure that is to have a turn x minus one multiplied in the numerator. So when X equals one, the whole function is equal to zero, and therefore it's an X intercept. Now, luckily through matching all of those conditions, we have also confirmed that that we have a ratio of quadrant questions. The numerator currently is a question of function. The denominator is a quadratic function S o. We have met all the conditions except for this last limit on this limit must equal to let's check to see if that is true currently in the form that it's written, the limit is X approaches negative one of this function alone. It's not equal to two on. We can see that for this function if these terms cancel X Plus one and X plus one. Uh, all this have expense one expense for evaluated acts and close the negative one because that's what the Limited's. So there's the middle equal negative one minus one. Think that one man is for or native to over. I think it's fine, which is equal to two or five. Our limit is not exactly equal to two years. It is exactly one that amount two. When we were one way that we concurrent thiss is to add a five to the front. This does not change any of the previous conditions. It is still the quadratic racial cryogenics, the vertical aspen tota. Still, that X equals four the X intercept X equal to one. And the function still has a removal. This continuity headaches and called negative one. The only difference is that here we would have five not to play by this term here. Right? Perhaps I have been appointed by making it one minus one. Here we would have tried multiplied by negative too. Canceling out what the denominator and overall Caviness A limited to when X equals negative one or what X approaches native one. And this is what we desired. So now we have a function that has made all the conditions, and this is exactly what the function have that's supposed to be. Now we're gonna answer parts A and B party. What is F zero alone? We just plug in zero into our function. Ah, simple. Fine. By canceling our X plus one allows it to be he's here to calculate time time zero minus one, divided by zero, minus four. This is far too laid one or negative five. Give it running before equal to five over for And that is our answer to party therapy. we're doing the limit as expertise. Infinity of dysfunction. Essentially, we're looking for whether his function as a horizontal asked me to. And again, if we use the produce form after cancelling out Texas ones, we should see that disfunction as expression. Infinity, the Quinn with student amore shape For away we divide each the numerator and the denominator. My ex, given its one minus or rex in the denominator one minus four works. And we know as X approaches infinity. These terms approaches hero. So this clearly shows a parliament approaches time as experts stupidity therefore and this is the answer to party. And we have found both party and B for the function that satisfied all the given conditions.

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