Epsilon Delta Proofs

[Johnmoon]

A little unsure about this one. I understand that you have to write the equation as an absolute value, but how is it simplified to x-1 if there's no x to start with?

 

[HallsofIvy]

The value of the function at x= 1 is 2. Further, the graph for x to the right of x= 1 is a connected curve so choosing a value close to 1 but larger than 1 will work. No, you do NOT have to write it as an absolute value. And you are to choose a value of x yourself.

 

[Dr.Peterson]

View attachment 12493
A little unsure about this one. I understand that you have to write the equation as an absolute value, but how is it simplified to x-1 if there's no x to start with?

There is no equation to write. You are supposed to "play" with the concept of a limit in the applet, by trying out some values of epsilon, and seeing for yourself whether there is a delta that "works". But you may want to start out by restating the definition of a limit with specific values in place, replacing x with 1 and L with 2. There's nothing more than that to do with the inequality.

Start with the epsilon shown; can you find a value of delta for which all points on the graph between the vertical yellow lines are within the horizontal lines? Then decrease epsilon, and see if you can still do it.

 

[JeffM]

View attachment 12493
A little unsure about this one. I understand that you have to write the equation as an absolute value, but how is it simplified to x-1 if there's no x to start with?

Let's try to dispel some confusion about terms.

A variable stands for a number that has not yet been specified. When you say "there's no x to start with" do you mean that there are no mental concepts called numbers or that 1 is not a numeric concept?

A function is a rule or algorithm that unambiguously specifies a unique number given any specific element in a set called the function's domain. (We could broaden the definition, but this will do for now.)

When we say that f(a) does not exist, we mean that a is not in the domain of f(x).

If the limit of f(x) at a exists, it is a number determined exclusively by the technical definition of limit, f(x), and a, but is completely independent of f(a). The independence of the limit of f(x) at a from f(a) is very important because frequently we are interested in that limit precisely because f(a) does not exist.

The limit of f(x) at a may not exist even though f(a) exists because the definition of limit is not met. That is the situation being illustrated here. f(1) = 2, but f(x) does not have a limit at 1.

The lmit of f(x) at a may exist even though f(a) does not exist. This is why we say 0 < |x - a| because f(a) is irrelevant to what we mean by a limit.

If the limit of f(x) at a and f(a) both exist, they may or may not be equal.

Now all of this is very nit picky. The functions that we deal with in algebra are continuous over most or all of their respective domains. f(x) is continuous at a if both the limit of f(x) at a and f(a) exist and are equal. The functions that we deal with in calculus are usually a sub-class of continuous functions called differentiable functions.

 

[Otis]

... When you say "there's no x to start with" do you mean that there are no mental concepts called numbers or that 1 is not a numeric concept?

My guess is that the OP thought they needed an explicit function definition (with x representing the input), and they wrote "no x to start with" because the exercise statement doesn't provide a definition.

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