sketch-the-graph-of-an-example-of-a-function-f-that-satisfies-all-of-the-given-conditions-f03-quad-l

Question

Sketch the graph of an example of a function $f$ that satisfies all of the given conditions.
$$f(0)=3, \quad \lim _{x \rightarrow 0^{-}} f(x)=4, \quad \lim _{x \rightarrow 0^{+}} f(x)=2$$ $$\lim _{x \rightarrow-\infty} f(x)=-\infty, \quad \lim _{x \rightarrow 4^{-}} f(x)=-\infty, \quad \lim _{x \rightarrow 4^{+}} f(x)=\infty$$ $$\lim _{x \rightarrow \infty} f(x)=3$$

Yaw Asomani · Accepted Answer

So how did this nice little sketch here? Eso straight away. What is what is the limit? As executions Negative for from the left right limit X approaches negative for right from the left. So from the left side. So inclusion approaching positive for from the left, right as X approaches positive for from the left. Right. So approaching positive four from the left is like this, right? You&#x27;re going like that because person before is here. Right? Separation from the letters like that. So if approaching 40 left, what are you gonna hit your hidden? This curve hit in this kind of here? Right? Which is sliding towards negative infinity. You see that? So is sliding toners negative infamy. The function is a current where you have to look at occurs and then what? ISS when you&#x27;re approaching four from the right right of a function. Still if approaching four from the right off the function right for this year. But when you&#x27;re pushing for from the right, you hit in this curve, right? And when you hit in discover of where is it? Sliding tours. Well, is it is rising up to infinity, right? As you continue to hit. Continue to get here. You hit this curve and it&#x27;s raining. Raising up to infinity. Right, So so rises to infinity right now. What is the limit? As you approach zero from the left, this is your right. So if I&#x27;m trying to approach zero from the left, what am I hidden? Hidden? Discard? Can you see that? And so where is this curve approaching? Well, it is approaching four. Right, seclusion for cause here is high. So it is approaching for right, cause for is gonna be here than three into, right? So as you continue to hit us function and this function is approaching for here, right, so and take out the air&#x27;s So this is for right now if you&#x27;re approaching zero from the right. Well, if a pollution zero from right, what is happening? So this is how you approach zero from the right? Right? So if you approach zero from right, then you hit this particular See the Citrus It dark here. I know that that is to write. Honesty is just not drawn to scale the discussion, but I just you right? So us You hit this zero here from the right hit us to start here, which is to write. So it is just by observation is very simple for you to look at it and then do it right so good. So once you have that, then look at the limit as you approach negative infinity. So the limit as you approach Negative infinity, This is negative. You go snow towards decide. Right? So as you are approaching negative infinity, what is happening to the function? The function is also going because you&#x27;re gonna as you go here on a result outside, the function is gonna hit this curve, right? And this car is sliding towards infinity. Can you see that it is a straight line slide? And tourists negative infinity, right? Because this is the negative side of why, right? So as you continue to go to infinity for this horizontal side, you hit this curve and in this curve is sliding towards infinity, Right? Deciding towards negative infinity to precise. So this is gonna be negative. Infinity! And what if your approach in, uh positive infinity So the limit as X approaches Positive infinity! So we&#x27;re approaching positive Infinity, what is happening to the function? But this is how you approach positive infinity, right for the horizontal side. What happens to the function? Well, you hit this curve as you proceed to positive infinity. You hit this curve, right? And when you heard that this current what is this curve? It is flapping out at three. Right. Here&#x27;s three here because 234 Right. So it is flattening out at three. Okay, so when you check, the curve is flattening out at three. Right? So this is just one example off the curve, right? You can have a so many examples. It&#x27;s possible, right? Particularly areas that I was using to do illustration. And you curve alone. So this is a curve, right?

Mary Wakumoto · Answer

Alright, here&#x27;s a fun one where we want to make a graph based on all these cool conditions, I&#x27;m actually gonna go backwards first. So let&#x27;s start off with rule six, we need f zero to be zero. So I&#x27;ll draw a nice dot at 00. So that one&#x27;s done now we need to know that as we approach infinity, then we have our function goes to zero, which means we have a horizontal ASM tote at y equals zero. So I&#x27;m just trying to draw that there. So we have a horizontal assam toad at y equals zero notice if I approach minus infinity, we also get zero. So therefore our horizontal as both of these give us the same results. So we&#x27;ve got that one done Now, we&#x27;re approaching, we&#x27;re doing number three now, so we&#x27;re approaching two from the left side and we get minus infinity. That means we have a a vertical as um toasts. Let&#x27;s go ahead and draw that. So at two we have a vertical ascent tote. So I&#x27;ll write that there. So that&#x27;s that x equals two. But as we approach um as we approach two from the left side, we&#x27;re going down to negative infinity. So I would say that um we are we could draw a graph, something like um uh this so we could draw something like this, we have a horizontal as some toad and then a vertical asthma tote says something like that. And then so we got that one done on the right side approaches plus infinity. We still have the horizontal assim tote. So being bound by both is typically what happens. So you can have more complicated ones, or it goes above and below. So for example, um I could even have this craft depending on the function, I could have something like uh this, ask them toe and then have it go over, whoops, I&#x27;ll make that a little bit better. Um Whoops! First of all, I just forgot that we have to go through our dot. So actually I&#x27;m glad I would like we thought it because we have to fix the fact that it must go through 00. We are bound by our ASM totes but we are having to go through 00, so I&#x27;m gonna have it go through 00 and then it&#x27;s gotta still be bound by the ASM totes. So there we go. Okay, so now we definitely, I almost forgot about number six, we&#x27;ve got number six, We&#x27;ve got our horizontal as went out for four and 5. We have our vertical ASM tote with a negative infinity on the left, positivity on the right. And so the only thing left is this one and I put this one separate because this one has issues. Um I want to make sure that the problem was written correctly because Um we can&#x27;t have a limit existing at two um when we have infinite limits and even if both of these both went to infinity approaching left and right at X equals two. Or both went to positive infinity officially. we wouldn&#x27;t write it as shown. So I think this one doesn&#x27;t fit the rest of the problem. Um But everything else, what I would basically say is limit X approaches to f f f X does not exist because left side and right side are not the same. The limit does not exist. And also if you approach infinity you can still say it does not exist. So anyway, hopefully that helped. Um let me just fix this bottom one because I started making the arrow point backwards on itself. So let&#x27;s get that looking just a little bit better. So let&#x27;s fix that real quick. So we&#x27;re gonna get rid of this little arrow peace. Okay, So we&#x27;re going to have it um go above and then through and then well, that&#x27;s supposed to go straight down. Okay? Anyway, hopefully that helped have a wonderful day.

Yaw Asomani · Answer

so have is, uh, sketch here. I can see that as x approaches to you having a vertical ass into. Right. So as you&#x27;re fruition to hear, you have any vertical vertical asking to eso take this one off. Yeah. So what&#x27;s your solution? Does to here You can see that A function a solution? Infinity Racer. The Lewit As X approaches to you can have a you know, it&#x27;s like is gonna continue on and on and on. It&#x27;s not drawn to scale is just a rough sketch, right? Eso if you&#x27;re trying to approach negative too from the left, right. If your fortune, if we&#x27;re approaching two from the left having apartment, not negativity. Yeah. No two. Exactly. If you&#x27;re approaching negative to from the left, Negative from the left. So fresh in negative from the left is like that, Right, cause this is from the left, right? You&#x27;re approaching negative two from values that are less than native to. Then as you&#x27;re approaching 92 you could hide it from us. Working is sliding too negative. And Finney, you see that from going to slide into negative infinity here. So as you&#x27;re approaching negative to the function is slide into negative infinity. Right? And what about as you&#x27;re approaching negritude from the right? Pretty negative from the right. Is this direction depression? Negative Two from values that are, uh, bigger than negative to say. So if you&#x27;re approaching negative too, then you have that, uh, vertical ascent is right. So as approaching native to from the right, what is happening less you as you approach native like this from the right like that. You know, you just your stopping point, right? That&#x27;s approaching there from the right. Then you can see that you have this thing here, go into infinity, right? Go symphony. Yeah. Good. So Ah, what is a functional value at zero. Right. So here, f at zero functional value and X equals zero, So zero. Right. So you have 00 as a coordinate. Right? This place is 00 here as a warning. Right. So All right. This one Probably. Yeah. Now, um, what is the limit? Asked you approaching Finney, right? As you approach infinity, what is happening to the functional values? So the limit as you approach negative infinity, you can see that it functions is approaching. This is negative. Infinity I&#x27;m approaching negative infinity and going Taurus. Just that site. Right? So negative five negative. 10 negative 1000 natives due to one million. So you go like that, right? You&#x27;re going like that, Affinity. What is happening to the function? Look at a function. The function is sliding. Told receiver, Right? It is almost touching this This horizontal axis here. Right, So the limit as X approaches. Negative. Infinity off the function zero. Right. Because as you move to the left side of functions, almost Russian zero. Right, That&#x27;s where you can see. And then what is for? No! Get a positive Infinity. The limits as you sly tours Positive infinity on the horizontal axis. Right. Well, who&#x27;s lad towards Infinity does this? This is the direction you gotta go. I look at this function here is also sliding. Almost tourist zero. It is almost hidden This horizontal axis where? Why is your right? So it is all also zero here, Right, So? So once you have that, you can see that this is just one example off a grand as satisfies all these conditions here, right?

Anjali Kurse · Answer

So what we&#x27;re gonna do here is sketch a graph of a function F that satisfies all of these conditions. So, the the limit ah the limit as X approaches to should be equal to negative infinity. Okay, then, the limit as X approaches infinity should be equal to infinity. The limit as X approaches Negative Infinity should be equal to zero. And the limit to more. The limit as X limit as X approaches zero from the right should be equal to infinity. and limit as X approaches zero from the left limit, as X approaches zero from the left should be equal to negative infinity, negative infinity. Okay, cool, so let&#x27;s draw a graph with all of those features. Let&#x27;s start at the ends of the graph. So as X goes off to positive infinity, which is over this way, it&#x27;s going to keep going. X gets bigger and bigger and bigger. This graph is going to go up to infinity. That looks like more of a line than a curve. Go up to infinity. Cool. Okay then, as X goes to negative infinity, it&#x27;s going to head to zero. Control it a little better. Zero. Okay. Or I guess either one of these could work. We&#x27;re gonna have to choose one. Okay, Now, as X approaches to to there&#x27;s going to be a vertical ascent owed and I know that because from both sides this graph at as X approaches to is going to be heading towards negative infinity just like that. Okay, now, how about has X ghost? So we&#x27;ve taken care of this one. This one this one. Okay. Now, as X approaches zero from the right, that&#x27;s what that little plus means. This is going to head up do positive infinity. Okay, Now, as X approaches zero from the left, X approaches zero from the left, we&#x27;re going to get negative infinity negative infinitely. Okay, now we&#x27;ve drawn all the portions we need so let&#x27;s just play connect the dots so this is going to come up that&#x27;s really awfully drawn. Okay like that. I&#x27;m going to head up deposit you know what, I&#x27;m just gonna redraw this part, that&#x27;s bothering me. Going to head up to Passo infinitely. I know it looks a little funny, sorry about that. Okay and now this one is going to head over to zero and look at that. We have fulfilled all the pieces of the graph. Let&#x27;s do a quick review. So as X approaches to from both the right and the left, this graph is headed down to negative negative infinity. That works. This is the line x equals two. Okay, now, as X approaches infinity, which is over this way the graph goes up to infinity, yep I could yeah that works okay. Now as X approaches negative infinity this graph is going to go to zero, Which it does heads over 20. The swimming Okay? And then As X approaches zero from the right, we got up to positive infinity here and zero from the left down to negative infinity here. So this red graph with some blacks and blues thrown on there for fun Z&#x27;s is what this graph is going to look like with all of these conditions satisfied.

Yaw Asomani · Answer

how This little sketch here? Eso I want to check a few things. You can see that the limits as X approaches to I limit s approaches to so as exits approaching this is to right. So as you&#x27;re approaching, uh, to you can see that the function is sliding tourist. Negative infinity. Right? So as you approach to here like that, the function here is sliding tours. Negative infinity, Because this is this is negative. Infinity this negative infinity for the y values. Right? So you&#x27;re sliding towards negative infinity. So Ah, this one here is sliding towards negative infinity. Well, the learned as X approaches zero from the left. So this is your right. So as you&#x27;re approaching zero from the ledges is how this is gonna be, right? So as you&#x27;re approaching zero from the left Musica Pollution zero from values that are lesser then, uh, zero. Right? So as you&#x27;re approaching zero from the left, where you hit you hit this infinite loop here that is going towards infinity. Can you see that for the y values? So here, this one is also negative. Infinity, right? Because you hit in here, continue to move forward towards you, then you sliding towards here for the white values, right? And then when you&#x27;re approaching is zero from the right. Right, So the limit as X approaches zero from the right off the function. Welcome. Pollutions air from the right there. You&#x27;re approaching like that, right? So if you&#x27;re approaching like that, then you&#x27;re gonna hit this curve, right? So, hitting that curve, then where are you going? You go into positive infinity. Why? Because, uh this curve here rises, tours positive infinity as you get closer to zero, right? This is getting close. Urges Europe as he gets close. Such as you know, you&#x27;re getting positive. Infinity, Can you see that? So what is happening here is this one goes to zero, right? Most of infinity goes to positive infamy because as you getting closer to zero like that, this one rises to infinity. Positive infinity, Right? Good. Now what is the limit? As X approaches Negative infinity. Well, this is how you approach negative infinity. You go this way, right? You&#x27;re going this way forever. So as you&#x27;re approaching negative infinity, what has happened? We could have a look at a function. What is happening to the function here. The function is approaching, uh, zero. Right functions approaching zero. So I ask you, the function is the current. That&#x27;s what I&#x27;m trying to see. So as you move here towards the left continues to be forever the curve. Here, look. A current fear. It is approaching the X axis here, which is zero. Right. So the function is approaching zero. Yeah, so it is just an example.

Ian Grigsby · Answer

All right, Y&#x27;all were doing graphs today. So we are supposed to sketch the graph of a function satisfying all six of the following criterion one, the limit as we approach zero from the left is to to when we approached zero from the right, we get 03 of the limit as we approach four from the left, we get three for the limit as we approach four from the right, we get zero and then five and six being f of zero equals two and f of four equals one. Now, as you can see, I&#x27;ve already laid out my graph nice to label the important point. So I&#x27;ve kind of label all the important Y coordinates and X coordinates. So the very first thing I want to take care of any time, I&#x27;m trying to graft something like this is any information that they give me about the original function in particular, F of zero equals two, and f of four equals one. So I better have points at zero comma two and four comma one. Now the limit information kind of tells me about what the function does around those points. Remember limits are giving me information about what the points around the point we&#x27;re caring about think we&#x27;re doing So for instance, as we approach zero from the left and are very first guy right here, ffx thinks that you&#x27;re going to land on to. So literally what this translates to is that as you walk up to X equals zero from the left hand side, hence a little negative right here, F of X says, man, you look like you&#x27;re approaching the y value of two so we can do kind of anything out here so long as we&#x27;re approaching the number to. So maybe something like that. Now the limit as we approach zero from the right hand side, hence the plus right here is zero. What that means is that ffx predicts as you walk up to zero from the right hand side, that you will land on the Y value of zero. So f of X might be doing something like this. Like Hey, we&#x27;re about to land on zero now we&#x27;re gonna do an open dot here. The reason we do an open dot is because we&#x27;re not technically ever touching zero comma zero, but we approach real close to it. Hence this whole limit that we get right here. Now when we move over to four, as we approach four from the left hand side yet again, that&#x27;s why we have that negative. Our function thinks that will land on the y value of three. So pick up where we left off and we go away up here yet again, an open dot remember limit means you get infinitely close to but you never touch that point. So we get that open dot. But notice also that when we approach four from the right hand side, meaning from the more positive side, our function thinks that we&#x27;re going to land on zero. So wait a second, we do something like this. Yeah. So yet again, important things in this problem. Make sure you first off graph these points, they have to be on your graph no matter what. That&#x27;s why we draw a dot at zero comma two and four comma one From there. We used our limit information as we approach zero from the left, we approached the y value of two as we approach zero from the right, we approached the Y value of zero. And then for four, as we approach four from the left, we approached the Y value of three. And as we approach four from the right, we approached the y value of zero, open dots anywhere that you didn&#x27;t happen. The land on a closed dot, that&#x27;s how you go and translate all this information into a graph and yours might have looked very different. By the way, there&#x27;s no reason that I have this like upward inflection right here. It could have been going downward. That would only be determined by more information given. But hey, that&#x27;s artist&#x27;s interpretation with what they gave me

Problem

Sketch the graph of an example of a function $f$ …

Problem 7 Medium Difficulty

Answer

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SEE GRAPH

Essential Calculus Early Transcendentals

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Anna Marie V.

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James StewartEssential Calculus Early Transcendentals

Grace H.

Anna Marie V.

Heather Z.

Kristen K.

James Stewart

Essential Calculus Early Transcendentals