Video: Finding the Limit at Infinity of a Rational Function

Find lim_(π‘₯ β†’ ∞) (5/π‘₯⁴ + 8).

03:53

Video Transcript

Find the limit of five over π‘₯ to the four plus eight, as π‘₯ approaches infinity.

We take π‘₯ approaching infinity to mean that π‘₯ increases without bound. So what happens as π‘₯ gets larger and larger? Well, for very large values of π‘₯, π‘₯ to the four is very very very very large. And so when you divide five by this very very very very large number, you get something very small. And so intuitively, as π‘₯ increases, five over π‘₯ to the four gets smaller and smaller, closer and closer to zero.

On the other hand, this eight doesn’t change. Its value stays the same. And so it’s reasonable to believe that the value of the limit of the whole thing is just eight. And indeed, this is true. We can use a graph of the function to show that as π‘₯ increases without bound, the value of the function approaches eight. We start with π‘₯ equals one. And we see that the value of the function, which is represented by the 𝑦-coordinate of a point on the graph, gets closer and closer to eight.

In fact, pretty soon, the graph of the function is indistinguishable from the line with equation 𝑦 equals eight. But in fact, if you could zoom in, you would be able to see that the graph of the function lies slightly above this line. In addition to perhaps the slightly handwavy intuitive method and the graphical method, there is an algebraic method which you might be more comfortable with, or feel is more formal.

We go back to the limit we want to evaluate. And we use the fact that the limit of a sum of functions is equal to the sum of the limits of the functions, assuming they both exist. And so we can break up the limit we want to evaluate into two limits. The limit of five over π‘₯ to the four, as π‘₯ approaches infinity and the limit of eight, as π‘₯ approaches infinity. This law holds not only when π‘Ž is some real number that π‘₯ is approaching, but also when π‘₯ approaches infinity or negative infinity.

And we can use another law of limits here to write the limit of a constant multiple of the function as that constant multiple of the limit of the function. The limit of five over π‘₯ to the four is therefore five times the limit of one over π‘₯ to the four. Perhaps it’s obvious to you right now that the value of the highlighted limit is just zero. But we can use another law of limits that the limit of a power of the function is just that power of the limit of the function.

And so we can write this limit in terms of the limit of the reciprocal function one over π‘₯, as π‘₯ approaches infinity. Here we also had to use the fact that one over π‘₯ to the four is one over π‘₯ to the four. We can’t continue this process forever, relating our limit, that we want to find the value of, to other limits, like the limit of the reciprocal function. Eventually, we’re going to have to evaluate some limits. We use the fact that the limit of the reciprocal function one over π‘₯, as π‘₯ approaches infinity, is zero. And the limit of the constant function eight, as π‘₯ approaches any value or indeed infinity, is just the constant eight.

You can take the values of these limits to be laws of limits in some sense. Hopefully, they are intuitively true. But they can also be proved rigorously using a definition of limits which you may or may not have seen already.

Our final step is just to evaluate. And we get the value eight, as before. You may have thought that the value of this limit was obvious from the start. If not, then hopefully, you pick up some of this intuition as you learn more. But it’s important to understand that in this case, our intuition can be justified by the laws of limits which can then be proved rigorously using a further definition.

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