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Sketch the graph of an example of a function $ f $ that satisfies all of the given conditions.

$ \displaystyle \lim_{x \to 2} f(x) = \infty $, $ \displaystyle \lim_{x \to 2^+} f(x) = \infty $, $ \displaystyle \lim_{x \to 2^-} f(x) = -\infty $, $ \displaystyle \lim_{x \to -\infty} f(x) = 0 $, $ \displaystyle \lim_{x \to \infty} f(x) = 0 $, $ f(0) = 0 $

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03:11

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Arthur S.

June 1, 2019

Hi great work!

Harvey Mudd College

Baylor University

Boston College

Lectures

04:40

In mathematics, the limit of a function is the value that the function gets very close to as the input approaches some value. Thus, it is referred to as the function value or output value.

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

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Sketch the graph of an exa…

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Sketch a graph of a functi…

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Alright, here's a fun one where we want to make a graph based on all these cool conditions, I'm actually gonna go backwards first. So let's start off with rule six, we need f zero to be zero. So I'll draw a nice dot at 00. So that one'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'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've got that one done Now, we're approaching, we're doing number three now, so we're approaching two from the left side and we get minus infinity. That means we have a a vertical as um toasts. Let's go ahead and draw that. So at two we have a vertical ascent tote. So I'll write that there. So that's that x equals two. But as we approach um as we approach two from the left side, we'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'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'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'm gonna have it go through 00 and then it'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've got number six, We'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'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't write it as shown. So I think this one doesn'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's get that looking just a little bit better. So let's fix that real quick. So we're gonna get rid of this little arrow peace. Okay, So we're going to have it um go above and then through and then well, that's supposed to go straight down. Okay? Anyway, hopefully that helped have a wonderful day.

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