Just starting calculus, noob question on limits

[JamesGold]

Hey guys, this forum has pretty much inspired me to start learning calculus, and, well, here I am.

So let's take this limit as an example of my point:

The limit of f(x) as x approaches 2 = 1.

(Pardon my lack of math symbols, I haven't learned how to use them yet, but I think this example is sufficiently simple so as to not merit their use)

What I don't understand is why we say f(x) = 1 as x approaches 2 when what's really happening is f(x) is approaching 1 as x approaches 2. As x is getting closer and closer to 2, f(x) is getting closer and closer to 1. At no point when x is close to 2 does f(x) equal 1.

Can someone explain this to me?

Thanks.

 

[Hurkyl]

What I don't understand is why we say f(x) = 1 as x approaches 2

We don't, unless we are being sloppy. The sentence you wrote is, after you remove all of the extra phrases:

The limit equals 1.​

Also, we do often say that f(x) approaches 1 as x approaches 2.

 

[JamesGold]

Ah, I see, I got the limit of f(x) confused with f(x) itself. Thanks, Hurkyl.

 

[Hurkyl]

It's an easy mistake to make -- in my estimation it's the most common cause of misunderstanding calculus.

It's not helped by the fact that two of the applications of a limit are:

1. You have a continuous function, you want to find the value f(2), and it so happens that this value is most conveniently computed by finding \lim_{x \to 2} f(x)
2. You are trying to define a function, and the easiest way to do so is to declare its values are equal to some limit -- e.g. one of the most straightforward ways to define the derivative is by the equation f'(x) = \lim_{y \to x} (f(y) - f(x)) / (y - x).

 

[Mark44]

The limit can exist whether or not the function is defined at a particular point. Using your example, it might be that \lim_{x \to 2} f(x) = 1 even though f is undefined at 2.

 

[DrRocket]

Hey guys, this forum has pretty much inspired me to start learning calculus, and, well, here I am.

So let's take this limit as an example of my point:

The limit of f(x) as x approaches 2 = 1.

(Pardon my lack of math symbols, I haven't learned how to use them yet, but I think this example is sufficiently simple so as to not merit their use)

What I don't understand is why we say f(x) = 1 as x approaches 2 when what's really happening is f(x) is approaching 1 as x approaches 2. As x is getting closer and closer to 2, f(x) is getting closer and closer to 1. At no point when x is close to 2 does f(x) equal 1.

Can someone explain this to me?

Thanks.

The important thing to remember in all of this is that the limit of f at 2 has nothing whatever to do with the value of f at 2, which may be different from f(2) or may not exist at all.

I will be to your benefit to spend the time to understand limits thoroughly. They are critical to an understanding of calculus.

 

[Angry Citizen]

The important thing to remember in all of this is that the limit of f at 2 has nothing whatever to do with the value of f at 2, which may be different from f(2) or may not exist at all.

Is it correct, then, to state that limits are a statement of continuity surrounding the output of the desired input?

 

[DrRocket]

Is it correct, then, to state that limits are a statement of continuity surrounding the output of the desired input?

Not really, but for a function that is continuous, or that has what is called a removable singularity that is one way to look at it.

The idea surrounding a limit is closely related to the notion of a cluster point. A cluster point of a set is one for which one can find points of the set arbitrarily close to it. So, a limit point of a sequence is a cluster point of that sequence.

Probably at the level of introductory calculus, the notion of a limit is best understood in terms of a limit of a sequence. Continuity of a function f at a point x is then simply the statement that the limit of f(x_n) is f(x) for every sequence x_n that converges to x. This notion can be extended to a more general setting, but it takes some topology to do that.

For your first exposure just get used to limits of sequences and functions. You can get fancier later after your intuition has developed a bit.