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Solving 1 Divided by Infinity

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

In this lesson, we will analyze the expression 1 divided by infinity through the use of limits, tables, and graphs. We will see how to work with the abstract concept of infinity within a mathematical expression.

Steps to Solve

We want to evaluate 1 divided by infinity. Infinity is a concept, not a number. We know we can approach infinity if we count higher and higher, but we can't ever actually reach it. Because of this, the expression 1/infinity is actually undefined, but that's not the end of the story! Otherwise, that would make for a very short lesson!

As we just said, we can approach infinity, so what we can do is look at what value 1/x approaches as x approaches infinity, or as x gets larger and larger. Let's use the following table to take a look at the value of 1/x for values of x as they approach infinity.

x 1/x
1 1
10 0.1
1,000 0.001
1,000,000 0.000001
1,000,000,000 0.000000001

Notice that as x gets larger and larger, approaching infinity, 1/x gets smaller and smaller and approaches 0. In mathematics, we call this a limit, and because we can't actually find a value for 1 divided by infinity, the limit of 1/x, as x approaches infinity, is as good as we're going to get.

In general, the limit of a function tells us what value the function approaches as x approaches a certain value, and we use the following notation for the limit of f(x) as x approaches a.


1divinf1


In this instance, we are taking the limit of the function 1/x, and x is approaching infinity.


1divinf2


From observing our table, we find that the limit of 1/x, as x approaches infinity, is 0.

Solution

Though the expression 1 divided by infinity is undefined, we can see what the expression 1/x approaches as x approaches infinity using limits. We found that the limit of 1/x as x approaches infinity is 0.


1dininf3


Graphs and Limits

In finding the limit of 1/x as x approaches infinity, we used a table of values to observe a pattern to evaluate the limit. This can also be observed graphically. On graphs, limits as x approaches infinity or negative infinity show up as horizontal asymptotes. An asymptote is a line that a graph approaches, but does not touch.

Let's think about our example for a moment. We found the limit of 1/x as x approaches infinity to be 0. Graphically speaking, this says that if we look at the graph of y = 1/x, we will see that as x gets larger and larger, or the further to the right that we go along the x-axis, the closer the graph will get to the line y = 0. However, again, since we can't actually reach infinity to evaluate 1 divided by infinity, the graph won't ever actually reach the line y = 0. It will just approach it. I don't know about you, but I think we just explained an asymptote!

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