Here, we’re going to formalize what you already know. If you understood the section on the precise definition of a limit, this section should be pretty simple. It’s another example of a delta-epsilon (that’s δ and ε) proof, this time for the case of x approaching infinity.
There are really 2 cases to deal with. I’ll state them at the beginning, then go through them separately.
Case 1: When the limit goes to some number L as x gets larger.
Case 2: When the limit goes to infinity as x gets larger.
The book divides the first case into right and left hand, but leaves that division of the second case for an exercise in the book. I think once you understand the limit as x goes to infinity, it’s pretty clear how it work as x goes to negative infinity. So, I’m just going to go over the two cases above in detail.
Here’s the book definition for case 1:
Let f be a function defined on some interval (a, ∞). Then,
means that for every ε > 0, there is a corresponding number N such that
if x > N, then |f(x) – L| < ε
OKAY, let’s unpack that.
First, you’ll notice there’s no delta. Why? Because in the definition of a limit, we were basically saying “if I take the input close to a certain value, the output approaches some certain value.” Now, we’re basically saying “if I take the input bigger and bigger and bigger, the output approaches some certain value.” So, we don’t need delta, which we originally used to indicate the small difference between the input and the value it was approaching.
Now then, let’s put the definition in layman’s terms. You’re saying you’ve got a function whose domain goes from some value a to infinity. A better way to say that would be “everything above a.”
Next, we say that as you take take that function’s input (x) larger and larger, the output, f(x) approaches a particular value. We call that value L (though we could just as easily call it D or P or Snorp or whatever).
Then we say that we can make x larger and larger, and that by doing so we’ll take the different between f(x) and L smaller and smaller.
I don’t want to linger here much more, because by now all this stuff should make sense. They go into the similar case of x getting infinitely negative. The only difference in the definition is that you change the “x > N” to “x < N.” The former is basically saying “go ahead, bitch. Pick any number. I can take x higher.” The latter is basically saying “go ahead, dickface. Pick any number. I can take x smaller.” And, as you do either of these things, bitchface, the distance between f(x) and L shrinks.
Now, the last little bit is the simplest limit definition you’ll see. Remember how we didn’t need delta, since we’re just taking x arbitrarily large? Well, now we don’t need epsilon, since we’re taking f(x) arbitrarily large. So, all we have left are M and N. In fact, this definition is so simple, it’s not really worth writing out rigorously. Here’s what it says: If the limit as x goes to infinity is infinity, that means that you can take f(x) arbitrarily large by taking x arbitrarily large.
Okay. We did it. We’ve gone through all the limits. It’s time to encounter the dreaded derivative.