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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 \infty} f(x) = 3 $, $ \displaystyle \lim_{x \to 2^-} f(x) = \infty $, $ \displaystyle \lim_{x \to 2^+} f(x) = -\infty $, $ f $ is odd

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Oregon State University

University of Michigan - Ann Arbor

Boston College

Okay, we want to graph the function F of X. You can see that I have the X axis, the F of X axis, I have a couple of key numbers to a negative 23 and negative three on the vertical axis to a negative 20 X axis. Uh Here's some restrictions that we have to make sure we meet. Uh As X approaches infinity, The function approaches three as X approaches to from the left side, the function heads towards infinity as X approaches to from the right side to function heads towards negative infinity. And then F is odd. Now, if is odd, we're going to address that last. We're going to graph the function on the right side of the Y axis scoops except Okay, so to address the F is odd, we're just going to graft ffx our function on the right side of the Y axis to vertical axis. Uh But then an odd function basically, um whatever you have in this part of quadrant, look at mirrored were reflected down in this quadrant. And whatever part of the great if you have in this quadrant will get reflected and mirrored uh in this quadrant over here. So we'll look at that last. Okay, uh as X is approaching two from the left side and from the right side, so we need a vertical assam toke line At x equals two. And because of the odd situation, we're gonna need another one at negative two, so we might as well draw that in now. So those red lines are vertical pass and talks and here's why we need it. Uh As X approaches to from the left side function heads up towards positive infinity. So it's X is approaching two from the left side, from the negative side to function heads up towards infinity. So let's grab this function in blue and uh let's have the function. There's nothing that says, the function can't be ah equal is zero when X zero. So let's put um this point on the graph of the function. Now as extra approaches to from the left side to function heads up towards positive infinity. And his extra approaches to from the right side to function heads down towards negative infinity. So his extra approaches to from the right side, the function is going to head down to negative infinity like this. But as X approaches infinity. So as we move further and further uh on the right side of the X axis. Uh as X approaches positive infinity, two functions approaching three. So we need a horizontal a sento uh at Y equals tree or f. Quebec's equals tree. So let's put a horizontal assam token here at three. And because of the odd situation, let's put another one at negative three. And just to make sure you can see that little negative sign, it got kind of covered up, let's put it in right there. So it's negative three right there. Okay, Now, as X is approaching two from the right side to function heads towards negative infinity. So X approaches to from the right side function heads down towards negative infinity, but as X approaches positive infinity as we move along the right side of the X axis to function approaches trade. So when we move in this direction, the function approaches the value of three. And then as we move towards to from the right side of to, the function goes down towards negativity. So we're looking at something that's going to get drawn like this. All right. So we've actually taken care of all three of these conditions. We just haven't taken care of the F's odd condition yet. All right. Let's see if we met all three of these accurately as X approaches infinity to function approaches three as we X approaches infinity As we move to the right, the function goes up towards three. Uh as X approaches to from the left side to function Heads up towards positive infinity. So as x approaches to from the left side, two functions going up towards infinity. As X approaches to from the right side to function goes down to negative infinity. As X approaches to from the right side, the functions going down towards negative infinity Now, uh nothing, there was nothing that said what had to happen to the graph. Um between zero and before you get too too, we just have X approaching two from the left side, but you know, from zero to like 1.5. Um there's nothing uh said about the function, so that's why we were at liberty to do whatever we wanted to uh for X equals zero. Um Up to, you know, not too close to too, but you know, save 1.5 1 and three quarters now. So we took care of all three of these conditions. Now we have to take care of the F is odd conditions. Um because F is odd, you're going to be reflecting uh bye in diagonal quadrants. So for example, this portion of the graph of F. A bex because I have an odd function, this portion that you see going up like this is going to be mirrored but down in this quadrant right here. So if we're going up like that, then we're going to be coming down like this. That's what an odd function does. Okay, it's going up like this in the first quadrant, so picture like a almost like a double reflection. Okay, it's going up like this in the first quadrant, so it's going down like that in the third quadrant. So we took care of this piece. Now we got to do the mirror of this as long is going to be a little trickier. Um This one as you went towards to was going down towards negative offending. So now as we go towards to we're going to be going up towards positive infinity. But then as ex moved to the right, the function approaches three Now, as we move to the left, the function is going to approach -3. So this little piece here is going to be like this let's see if we can pull this off approaching negative too. We're gonna go up towards infinity. So let's try to be uh about evenly spaced as we were here from to this little X intercepted two. We want kind of like this similar distance over here, uh from negative to okay, so this little distance here should be about the same, should be the same distance as here. Uh This curve going up now, you're going to have that part coming down and then this part that was going down like that is going to continue up like this. So, here is an example of an odd function F a bex that satisfies these other conditions.