Limits at Infinity

With limits, we can look at how a function behaves as values approach infinity. When we look at this behavior, it tells us information about a function’s asymptotes.  

Definition: Let f be a function that is defined on an interval (a,\infty). Then, lim_{x \to \infty}f(x) = L means that values of f(x) can be made close to L by taking large x values.

This idea is referred to as a horizontal asymptote. Another way to consider it is that no matter how large x grows, f(x) will never go beyond the value L, it will only approach it. For an example, consider the function f(x) = 2^{-x}

As you can see, this function gets very close to 0, almost seeming to equal 0 as x grows larger. However, there is no power that makes f(x) = 2^{-x} equal to 0, so f(x) will never actually equal 0, just approach it as x approaches infinity. We can say from that that f(x) has a horizontal asymptote at x = 0.

We can similarly consider what happens when we input extremely small values, or rather, approach negative infinity.

Definition: Let f be a function that is defined on (-\infty,a). Then, lim_{x \to -\infty}f(x) = L means that values of f(x) can be made close to L by taking x as a sufficently large negative number.

If we look at the reflection of f(x) = 2^{-x}, which is f(x) = 2^x, we can see this more clearly.

In this instance, it looks that as f(x) continues in the negative direction, the function continues to approach 0. Again, it can never equal zero, but it continues to grow closer and closer as we approach negative infinity. This too can be considered a horizontal asymptote.

Definition: L is considered a horizontal asymptote of f(x) if lim_{x \to -\infty}f(x) = L or lim_{x \to \infty}f(x) = L

In a lot of cases, these limits may not approach an actually value L, but rather approach \infty or -\infty themselves. These cases tell us that the function will continue to grow infinitely, so there is no asymptote to worry about. This gives us good information on the domain of functions, as well as how the functions actually look if we need to graph them.

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