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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 $,

$$\begin{array}{ll}

\lim _{x \rightarrow 2} f(x)=-\infty, & \lim _{x \rightarrow \infty} f(x)=\infty \\

\lim _{x \rightarrow-\infty} f(x)=0, & \lim _{x \rightarrow 0^{+}} f(x)=\infty \\

\lim _{x \rightarrow 0^{-}} f(x)=-\infty &

\end{array}$$

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Oregon State University

Harvey Mudd College

University of Michigan - Ann Arbor

Boston College

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.

The University of Alabama