Need some help with the how...

[CLSE]

Hi all, I'm new on this forum.

I'm taking Calc I for an engineering course.

I am familiar with Calc and what it is used for (and I really love doing all types of math), but the problem that I have consistently run into with math courses is a combination of math-speak and poor description of what is being done and more importantly, how it's being done.

The math-speak might as well be Greek (no offense to the mathematicians), it makes no sense to me - show me how I can apply it in a process and I can then figure out what it's doing, if that makes any sense.

Something is not clicking for me and it's frustrating.

As an example - From my Calc book - Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if lim x-->c- f(x)=L and lim x-->c+ f(x)=L.

I understand that I'm approaching the limits from the right and left and that if both functions equal L when x is at c then I've reached a limit, but what does that actually tell me and what do I do with it?

If x goes past c, have I exceeded the limit, or is it irrelevant in this case and if so, why?

Normal thinking would be that when you come to a limit, you've reached the end of the road, but then you have infinite limits - no ending.

So, with an infinite limit, have you really just identified that there is no limit, or is it something else?

I hope this makes some sense.

Thanks in advance.

 

[Anonymous1]

Here is an example of "why."

---0-------\(\displaystyle f(x)\)
\(\displaystyle \rightarrow \\\\ \\\\ \leftarrow\)

\(\displaystyle f(x)\) is a function that has a "hole" at some point. We want to fill it in, but first we have to make sure the discontinuity is "removable." So we approach it from both sides until we are infinitely close on either side. Then we color it in.

An unremovable discontinuity, would be something like:

------______

 

[undefined]

Hi all, I'm new on this forum.

I'm taking Calc I for an engineering course.

I am familiar with Calc and what it is used for (and I really love doing all types of math), but the problem that I have consistently run into with math courses is a combination of math-speak and poor description of what is being done and more importantly, how it's being done.

The math-speak might as well be Greek (no offense to the mathematicians), it makes no sense to me - show me how I can apply it in a process and I can then figure out what it's doing, if that makes any sense.

Something is not clicking for me and it's frustrating.

As an example - From my Calc book - Let f be a function and let c and L be real numbers. The limit of f(x) as x approaches c is L if and only if lim x-->c- f(x)=L and lim x-->c+ f(x)=L.

I understand that I'm approaching the limits from the right and left and that if both functions equal L when x is at c then I've reached a limit, but what does that actually tell me and what do I do with it?

If x goes past c, have I exceeded the limit, or is it irrelevant in this case and if so, why?

Normal thinking would be that when you come to a limit, you've reached the end of the road, but then you have infinite limits - no ending.

So, with an infinite limit, have you really just identified that there is no limit, or is it something else?

I hope this makes some sense.

Thanks in advance.

Hmm, usually definitions are chosen carefully so as to avoid ambiguity and be precise, but sometimes it may take some time to decipher the formal language. Also, there may be more than one way to define a term, or to express a certain definition.

The definition you wrote for limit is, in fact, practical, at least in a certain sense. From this definition you can tell at a glance that, for the below figure, the limit of f(x) as x approaches c does not exist.

Clearly, the right and left limits exist and do not equal each other, therefore the limit does not exist.

Like Anonymous1 mentioned, this type of reasoning is helpful when it comes to telling whether functions are continuous at certain points or on certain intervals.

I do not have a good example of how this would be useful in an engineering scenario.

Limits basically deal with what happens when x approaches a given value, or when x approaches positive or negative infinity. Thus, we will never "go past c," and when we discuss limits we are not thinking about a point beyond which f(x) becomes undefined (in general).

Here's an example that people often have trouble accepting when first introduced to decimals. Why does 0.9999... = 1? The answer is with limits.

0.9999... = 9/10 + 9/100 + 9/1000 + ...

This is an infinite series, and the limit of the sum as the number of terms approaches infinity is 1. We say that the series converges to 1, and they are equal.

I hope you can find good materials/professors/etc. who can emphasize the practical applications. I happen to enjoy the theory.

 

[Anonymous1]

Speaking about \(\displaystyle \infty\) is always tricky.

Examine this statement:

\(\displaystyle \text{"As x gets really big}\) \(\displaystyle f(x)=\frac{1}{x} \text{ gets really small."}\)

Do you agree? Plug \(\displaystyle f(10^{10})\) into your calculator and see what you get. Technically speaking \(\displaystyle f(x)\) is never \(\displaystyle 0,\) and you probably realize that \(\displaystyle f(x)\) will never get "past" \(\displaystyle 0;\) only closer and closer.

"Limit" is the name of this value that we get really close to. Or in "math speak:"

\(\displaystyle \lim_{x\to\infty}\frac{1}{x} = 0.\)

 

[CLSE]

Anonymous1, undefined,

That actually helped out, many thanks -

So, I'm actually dealing with different "limits", each of which are not the same thing - it may seem like semantics, but that's part of what was hanging me up.

So, I am determining what the left and right limits are, independent of anything else.

If both the left and right limits reach a point (x,y) where they are equal to each other, I have reached "the" limit".

From a trig-ish perspective, the left and right limits would be (P-ΔP)+P and (P+ΔP)-P, respectively (P would actually be unknown in this case, I'm just using it to keep it clear in my head).

As I make ΔP smaller and smaller, if the two reach a point where they are equal to each other, then I have reached the point P, which is "the" limit.

If they never reach each other, then I have a discontinuity or the left and right limits are infinite.

About infinite limits, using your example, Anonymous1 -
if I had lim (-x)->0 -(1/x) on the left side and lim x->0 1/x on the right side, then f(x) would become infinitely close to 0, but never actually reach 0, in which case, "the" limit doesn't exist.

undefined, here's an contrast of two ways of explaining a derivative -

First, the mathematical definition -
"the derivative of f at x is given by f'(x) = limΔx->0 f(x+Δx)-(fx)/Δx provided the limit exists. For all x for which this limit exists, f' is a function of x."

Then, the explanation from an engineering formulas book that I have (it doesn't cover limits, it goes straight to derivatives) -
"Where Δx is infinitely small, i.e., where Δx approaches zero, the slope at P becomes the limiting value of the slope of one of the secants. This slope is the "derivative" or "differential coefficient" of the function at P."

The second one makes more sense to me - the first one tells me what a derivative is, but with the second one, and like you said, the first one may require some deciphering, but for the second one, I can visualize what it should "look" like.

I think a combination of the two approaches works for me, though - I'm looking to see if the left and right limits meet at P - if they do, P is "the" limit and I then determine the slope at P, which is the derivative.

You guys rock, many thanks.