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

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02:50

Daniel Jaimes

Calculus 1 / AB

Chapter 2

Limits and Derivatives

Section 6

Limits at Infinity: Horizontal Asymptotes

Limits

Derivatives

Harvey Mudd College

University of Nottingham

Idaho State 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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So what we'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's draw a graph with all of those features. Let's start at the ends of the graph. So as X goes off to positive infinity, which is over this way, it'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's going to head to zero. Control it a little better. Zero. Okay. Or I guess either one of these could work. We're gonna have to choose one. Okay, Now, as X approaches to to there'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've taken care of this one. This one this one. Okay. Now, as X approaches zero from the right, that'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're going to get negative infinity negative infinitely. Okay, now we've drawn all the portions we need so let's just play connect the dots so this is going to come up that's really awfully drawn. Okay like that. I'm going to head up deposit you know what, I'm just gonna redraw this part, that'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'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's is what this graph is going to look like with all of these conditions satisfied.

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