Understanding Limits in Calculus: Exploring the Definition and Misconceptions

[ksinelli]

I am trying to get a head start and learn some calculus before my class begins this fall. I'm trying to learn from Khan Academy, but I'm already confused. I thought the definition of a limit was a value that could never be reached, though could be infinitely close to being reached. Here is a problem I just encountered in practice.

http://s16.postimg.org/qy9d2bxed/limit.jpg

I initially chose that a limit for this function does not exist, but apparently 1 is the correct answer. But I am confused because 1 is a point that can actually be reached. It can be reached when x = 1. So this makes me wonder... couldn't I do the same thing for any other point on the graph? Couldn't I say that the limit of f(x) as x approaches 2 = -5? And for that matter, wouldn't that mean I could say the same thing for any point on any parabola similar to this?

 

[ksinelli]

This website does the same thing. They introduce a limit as some value that can never quite be reached, but can get really really close. And then by the end of the introduction, they are suddenly saying that even if it can be reached, as long as it the same number when being approached from the right and left side, it is still a limit. (specifically the second to last example where they say the lim f(x) as x approaches 0 = 2. From looking at the graph, it seems pretty clear that 2 can in fact be reached when x is = 0.)

http://www.coolmath.com/lesson-whats-a-limit-1.htm

 

[johnqwertyful]

A limit is completely irrelevant to what happens at the point. The point could be 1, -345353, pi, 4.55. The limit will still be the same. The key is that a limit is what happens "around" the point, not what happens AT the point.

Also, a little bit of a technical detail, irrelevant to you now is that "infinitely close" doesn't really mean anything. The idea is "arbitrarily close". To say that lim x->c f(x)=L is to say that by making x arbitrarily close to c, you can make f(x) arbitrarily close to L. "Infinitely close" isn't well defined.

The key here to your question though is the idea of continuity. A function is CONTINUOUS at a point if lim x->c f(x)=f(c). In other words, if the value of the limit is the value of the function at that point. It's a common fact in calculus that ALL polynomials are continuous. In other words, if you're ever presented with a limit of a polynomial (a parabola is a special case of a polynomial), you can just plug in the point and you'll get the answer.

 

[johnqwertyful]

Other continuous (everywhere) functions are sin, cos, exponential functions.

lim x->c sin(x)=sin(c) always
lim x->c e^x=e^c always

 

[johnqwertyful]

A video that can explain it clearer than I can.

 

[Stephen Tashi]

This website does the same thing. They introduce a limit as some value that can never quite be reached, but can get really really close.]

Read the actual definition of $\lim_{x \rightarrow a } f(x) = L$. There is nothing in that definition about "reaching" or "not reaching" anything. Putting definitions into your own words is recommended to students of the liberal arts, but it usually doesn't work in mathematics. Mathematical definitions mean what they say.

If you are reading material that presents a mathematical concept in a non-technical and informal way then you can't expect it to have strict logical consistency.

 

[verty]

Also, a little bit of a technical detail, irrelevant to you now is that "infinitely close" doesn't really mean anything. The idea is "arbitrarily close".

Infinitely close and arbitrarily close are saying the same thing: there is no gap between the limiting value and the function's values. This is a useful way to think of it, I think. I agree with Simon Tashi that words are vague and not as precise as strict logical symbols. Learning logic is most useful for understanding exactly what the definitions are saying.

 

[johnqwertyful]

Infinitely close and arbitrarily close are saying the same thing: there is no gap between the limiting value and the function's values. This is a useful way to think of it, I think. I agree with Simon Tashi that words are vague and not as precise as strict logical symbols. Learning logic is most useful for understanding exactly what the definitions are saying.

I'd say it's different.

By choosing delta small enough, you can make f(x) within epsilon of L, for all epsilon>0. That means that you can make f(x) arbitrarily close to L. I don't know what "infinitely close" would even mean. If you say that it means the same as arbitrarily close, that seems confusing. Because when I hear "infinitely close", an image of a point that is "the closest" to L comes up, and that's not possible, as I'm sure you know.

Maybe they can mean the same thing, but I think "arbitrary" is a much clearer word.

 

[verty]

I'd say it's different.

By choosing delta small enough, you can make f(x) within epsilon of L, for all epsilon>0. That means that you can make f(x) arbitrarily close to L. I don't know what "infinitely close" would even mean. If you say that it means the same as arbitrarily close, that seems confusing. Because when I hear "infinitely close", an image of a point that is "the closest" to L comes up, and that's not possible, as I'm sure you know.

Maybe they can mean the same thing, but I think "arbitrary" is a much clearer word.

Sorry, I see the problem now. If we consider that the natural numbers get bigger and bigger without bound, we could say they are infinite in size, but this usually means that each is infinitely large, this is a confusing use of words. With "infinitely close", the same happens. Points get closer and closer, but to say they get infinitely close is sure to be confusing.