The reciprocal function?

Considering the parent reciprocal function, explain what happens to the value of y as x approaches 0 from the left side. Explain, graphically and algebraically, why this occurs. Give as much evidence as possible.  

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1 Answer

  • 2 months ago
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    Okay, let's look at that.

    Let's take *negative* values of x that get closer to zero and calculate the reciprocal.

    x= -5, y= -1/5 (-0.2)

    x= -4, y= -1/4 (-0.25)

    x= -3, y= -1/3 (-0.333...)

    x= -2, y= -1/2 (-0.5)

    x= -1, y= -1

    x= -0.1, y= -10

    x= -0.01, y= -100

    x= -0.001, y= -1000

    x= -0.0001, y= -10000

    What do you notice as you get closer and closer to x=0 from the left? What is happening to the y value? Seems to be getting more and more negative without bound. It looks like it would continue forever toward negative infinity.

    Now look at the graph and see if you see the same thing.

    As you move along the graph starting at around -5 and moving *right* you get a slightly negative value of y. But as you move closer and closer to zero, it starts going down, lower and lower. As you get ever closer to x=0, the value of y goes down towards negative infinity.

    What can you conclude? What would you do to show this more "algebraically"? Do you understand it from the graph?

    Once you've done that, you might want to explore what happens when you approach x=0 but from the *right*. Here you have positive values of x getting closer to zero. What happens to the value of y? Is it similar? Is it also going towards *negative infinity* also or perhaps in a different direction?

    If the limit is different from either side can you say it has a single limit as x approaches 0 or not?

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