This is something I blogged about a couple years back (yes, it's not 100% accurate/applicable, but it's good enough):
I’ve been reviewing calculus online lately, currently doing limits specifically. Overall, the concept of the limit is easy to grasp intuitively. But mathematically, it can be quite challenging to understand the precise definition of the limit (I’m referring to the epsilon-delta definition of the limit). And I’ve been struggling the whole day trying to really understand what the definition is actually saying. I think I finally get it now, but let me see if I can explain it in writing here on this page.
So the definition goes like this:
Have function f(x) defined on an interval containing x=a, except possibly at x=a. The limit of f(x), as x approaches a, is equal to L means that for every “epsilon” (that is greater than zero), there is a “delta” (also greater than zero), such that |f(x) – L| < “epsilon” whenever zero < |x – a| < “delta”.
Confusing, yeah? Don’t feel bad if you do find this confusing. Apparently, even calculus graduates struggle with really understanding the definition.
Anyway, let’s see how to go about explaining the definition:
But first: I am going to assume you have adequate knowledge of functions and limits and such, and your main struggle in the topic of limits is to do with understanding the definition above. If you have no idea what limits are, then this is going to be a difficult read for you regardless of my explanation.
With that notice out of the way, let’s start the explanation.
Say I am convinced a limit exists for a certain function f(x) at x=a, with the limit called L. The function, by the way, is defined at every point on a certain interval (except perhaps at x=a). So I know there is a limit, and I know what it is. But how can I prove that I know the limit is L?
I can prove it as follows:
I challenge my knowledge with every possible value of “epsilon” that is greater than zero. Each value of “epsilon” represents a y-distance away from L (on either side of L). On the other hand, each value of “delta” represents an x-distance away from x=a, the x-coordinate that corresponds to L (on either side of x=a).
The challenge is to come up with a corresponding value of “delta” (in the case of each value of “epsilon”) that consistently keeps the portion of the graph within the x-distance of “delta” from x=a and within the y-distance of “epsilon” from L. The challenge fails if both aforementioned conditions are not met. The smaller the “epsilon”, the more challenging it is, since a small “epsilon” means a small distance away from L.
If, through logical thinking, you are able to deduce that every possible “epsilon” value has a corresponding “delta” value that works, then this means that no matter how close you want to get to the limit, there will always be a portion of the graph that will consistently stay within both the “delta” distance chosen and the “epsilon” distance it corresponds to. This shows that the graph is indeed approaching L as x approaches a, which means L is indeed the limit as x -> a.
And I think I’m done. If not, I’m still done anyway, because I’m tired.
I’ve been reviewing calculus online lately, currently doing limits specifically. Overall, the concept of the limit is easy to grasp intuitively. But mathematically, it can be quite challenging to understand the precise definition of the limit (I’m referring to the epsilon-delta definition of the limit). And I’ve been struggling the whole day trying to really understand what the definition is actually saying. I think I finally get it now, but let me see if I can explain it in writing here on this page.
So the definition goes like this:
Have function f(x) defined on an interval containing x=a, except possibly at x=a. The limit of f(x), as x approaches a, is equal to L means that for every “epsilon” (that is greater than zero), there is a “delta” (also greater than zero), such that |f(x) – L| < “epsilon” whenever zero < |x – a| < “delta”.
Confusing, yeah? Don’t feel bad if you do find this confusing. Apparently, even calculus graduates struggle with really understanding the definition.
Anyway, let’s see how to go about explaining the definition:
But first: I am going to assume you have adequate knowledge of functions and limits and such, and your main struggle in the topic of limits is to do with understanding the definition above. If you have no idea what limits are, then this is going to be a difficult read for you regardless of my explanation.
With that notice out of the way, let’s start the explanation.
Say I am convinced a limit exists for a certain function f(x) at x=a, with the limit called L. The function, by the way, is defined at every point on a certain interval (except perhaps at x=a). So I know there is a limit, and I know what it is. But how can I prove that I know the limit is L?
I can prove it as follows:
I challenge my knowledge with every possible value of “epsilon” that is greater than zero. Each value of “epsilon” represents a y-distance away from L (on either side of L). On the other hand, each value of “delta” represents an x-distance away from x=a, the x-coordinate that corresponds to L (on either side of x=a).
The challenge is to come up with a corresponding value of “delta” (in the case of each value of “epsilon”) that consistently keeps the portion of the graph within the x-distance of “delta” from x=a and within the y-distance of “epsilon” from L. The challenge fails if both aforementioned conditions are not met. The smaller the “epsilon”, the more challenging it is, since a small “epsilon” means a small distance away from L.
If, through logical thinking, you are able to deduce that every possible “epsilon” value has a corresponding “delta” value that works, then this means that no matter how close you want to get to the limit, there will always be a portion of the graph that will consistently stay within both the “delta” distance chosen and the “epsilon” distance it corresponds to. This shows that the graph is indeed approaching L as x approaches a, which means L is indeed the limit as x -> a.
And I think I’m done. If not, I’m still done anyway, because I’m tired.
