### Video Transcript

Consider the graph of the function π of π₯ is equal to one divided by one minus π₯ plus two. What is the end behavior of the graph as π₯ approaches one? Option (A) the value of π¦ approaches β when π₯ gets closer to one from the positive direction and approaches negative β when π₯ gets closer to one from the negative direction. Option (B) the value of π¦ approaches negative β when π₯ gets closer to one from the positive direction and approaches β when π₯ gets closer to one from the negative direction. (C) The value of π¦ approaches β when π₯ gets closer to one from the negative direction or from the positive direction. Or is it option (D) the value of π¦ approaches negative β when π₯ gets closer to one from the negative direction or the positive direction?

In this question, weβre given the graph of a function: π of π₯ is equal to one divided by one minus π₯ plus two. And we need to use this graph or the given function to determine the end behavior of this graph as the values of π₯ approach one. And thereβs many different ways we could go about answering this question. For example, we can determine the end behavior of this graph as π₯ approaches one entirely by using the given function.

To see how we might do this, letβs consider the denominator in the given function: one minus π₯. If our values of π₯ are greater than one, which means π₯ is in the positive direction of one, then we can note one minus π₯ will be negative, since weβre subtracting a number bigger than one from one. We can also note as the values of π₯ get closer and closer to one, the magnitude of one minus π₯ will get closer and closer to zero. In other words, the distance between one and π₯ gets smaller as π₯ gets closer and closer to one. So weβre dividing one by a negative number with a very small magnitude. And as the magnitude of this number gets smaller, its reciprocal will get larger in magnitude. And adding two to this value wonβt change this fact. So π of π₯ will approach negative β as our values of π₯ approach one from positive direction.

However, this level of analysis is very difficult. So, instead, itβs very useful to be able to do this from a given diagram or, if possible, to sketch a diagram to help us determine this information. We can do this by recalling the input values of the function are the π₯-coordinates of the point on the curve and the output values are the corresponding π¦-coordinates. This means we can determine what happens to the outputs of the function as our values of π₯ approach one from the positive direction by seeing what happens to the π¦-coordinates of points which lie on the curve.

To do this, letβs add in the vertical line π₯ is equal to one onto our diagram. We can see that the curve gets closer and closer to this line. However, it never touches the line. This is a vertical asymptote of the curve. In particular, we can see as the values of π₯ approach one from the right, the π¦-coordinates of the points on the curve are going lower and lower. Theyβre unbounded, so theyβre approaching negative β. And we get a similar story if we look at the values of π₯ approaching one from the negative direction. We can see that the π¦-coordinates of the point on the curve are getting larger and larger, and theyβre unbounded. So theyβre approaching positive β.

And we can see that this only matches option (B). The value of π¦ approaches negative β when π₯ gets closer to one from the positive direction and approaches β when π₯ gets closer to one from the negative direction.