# How to explain limits?

1. Jan 28, 2008

### duki

Hi all,

Does anyone know of a couple of good links I could view that would explain limits?

2. Jan 28, 2008

### arildno

3. Jan 28, 2008

### duki

Ok Great. I'll try to tell you what I understand and what I don't... sorry if it gets lengthy.

First off, I'm not sure what is even meant by "finding the limit". I'm not exactly a visual learner, but I do like to know what the equations are showing. Why is it called a limit?

One example of a problem I don't quite get is this:
$$lim as x->3 |x-3|/x-3$$
The notes I have are it's und. at x = 3, x > 3 = +1, x < 3 = -1 and the ultimate answer is no limit. Why (especially the part about no limit)? Are these simple concepts that I shouldn't be having difficulties with?

We have gone over the basic limits:

1. $$lim b = b as x->constant$$
2. $$lim x = c as x->constant$$
3. $$lim x^n = c^n as x->c$$

What do these mean? Where do the variables come from?

An example problem in my notes is as follows:

$$lim as x-> x^4 - x^3 + x - 1/ x^3 + 4x - 5$$
so then you divide by x-1?? Not sure why you do that.
so after dividing you use synthetic division and get $$x^3 + 1$$
apparently this is the final answer, but again... what does this show?

My main question I suppose is, when looking at a limit equation, how do you know where to start?

Many kudos for your help guys. :) Thanks!
ps) I'm not sure that I'm using the tex tags correctly.. sorry if I messed up.

4. Jan 28, 2008

### VietDao29

w00t, I really have hard time reading all your LaTeX's, and figuring out what they mean. :((

You can learn LaTeX here. There are 3 .pdf's files there to download. It's about some pages long, and will cover most of the LaTeX basis for you.

You can always click on another user's LaTeX image, and see its code.

Roughly speaking, limit is the value that some function tends to, when its parameter tends to some other value.

Like $$\lim_{x \rightarrow 2} x ^ 2 = 4$$. That means, when x is "close" to 2, x2 is "close" to 4.

The definition of limit, say: $$\lim_{x \rightarrow x_0} f(x) = L$$ if and only if
$$\forall \epsilon > 0, \exists \delta > 0 : 0 < |x - x_0| < \delta \Rightarrow |f(x) - L| < \epsilon$$

That means, no matter how "close" you want f(x) and L to be, there'll always be a neighborhood of x0, not containing x0 itself, (i.e, f can be undefined at x0, but still have limit at it), in which the value of f(x), and L are as "close" as you wish.

Using the definition above, one can prove that the limit is unique, independent of how the variable tends to a given value (i.e, from the left, the right, fast, or slow). You can find this proof in your book. It's pretty simple. :)

In the example above, there are 2 limits (one for the left, and one for the right), so the limit does not exist (since it does not satisfy the definition of limit).

For (1), if f(x) = c, where c is a constant, i.e f(x) is a constant function. Then for every value b, we have:

$$\lim_{x \rightarrow b} f(x) = c$$ (you can use the definition to prove this, by choosing any $$\delta > 0$$, like $$\delta = 1 ; \ 2 ; \ 3 ; ... ; \ 0.5$$).

I'm not sure what you means in (2).

You can prove (3) using the definition. The limit means that xn is close to cn, when x is close to c.

What value does x tends to?

There are some Indeterminate Forms of limit: $$\frac{0}{0} ; \ \frac{\infty}{\infty} ; \ \infty - \infty ; \ 0 ^ 0 ; \ 1 ^ \infty ; \infty ^ 0$$

Each form has its own way to solve.

Like, when you see the Indeterminate Form 0 / 0, it's common that you'll factor the numerator, and denominator, and simplify the expression by cancellation.

Well, book is your friend, I think you should re-read the book again. Or you can search for Limit on the Internet, like on wiki: Limit of a function.

Hope it helps. :)

Last edited: Jan 28, 2008
5. Jan 28, 2008

### Feldoh

Basically the idea of a limit is to be able to find values of functions of functions that would otherwise be impossible to calculate. For example in calculus we take the average slope between two point, then take the limit as the change in the points to be infinitely small. Without the concept of the limit we'd get a slope of some number over 0. However the limit allows us to APPROACH 0, but never reach it in a sense.

When you find the limit of a function you're essentially finding the value the function APPROACHES as x -> somenumber -- The idea of approaching some number is important when dealing with holes and asymptotes, that is why we say "approach".

In order for the limit $$\lim_{x\rightarrow a} f(x)=L$$ to exist:

$$\lim_{x\rightarrow a^+} f(x) = \lim_{x\rightarrow a^-} f(x) = L$$

Essentially the left and right-hand limits must be equal. The concept of a left handed and right handed limit is that you approach the function from either the left of right side of the point in question (-) is left-hand limit, (+) is right-hand limit.

In the case $$f(x) = \frac{3|x-3|}{x-3}$$

$$\lim_{x\rightarrow 3^-} f(x) = -3$$

$$\lim_{x\rightarrow 3^+} f(x) = 3$$

There for the limit itself doesn't exist. This is because there is a break in the graph at x=3, if you graph it you will see this.

$$\lim_{x\rightarrow a} b = b$$ just means that the limit of a constant will be that constant (since it is, you know, constant). The same applies to your second stantment?

$$\lim_{x\rightarrow c} x^n = c^n$$ -- If this is the case this means that the function is continuous at x=c (no holes, breaks, etc.) This just means as x gets closer to c the function gets closer to c^n, where n is probably taken to be some constant.

As for your last problem I'm not really sure what you're trying to say.

When you start a limit problem, the first thing you want to do is determine if the limit exists. If it does then you want to find a way to solve it such that you get a valid answer.

6. Jan 30, 2008

### duki

Thanks a bunch for the help guys.
We're now working on problems with geometric functions:

$$lim x->3 tan(pi(x)/4) = -1$$
but why?

7. Jan 31, 2008

### VietDao29

Your LaTeX looks horrible.. :yuck: Didn't you have a quick glance at the link I've given? =.=" It didn't seem to improve at all since your last post!!

Once you decided to use LaTeX, you should/must learn, and use it properly, or the outcome will be super-ugly. Your LaTeX above is a fine example of it. Just look at your LaTeX again, do you really know what it is saying?!?! If even you can't, then what are the chances of us all understanding your LaTeX?!?!

Or, you can even simply type it out normally, like this:

lim tan(pi(x) / 4)
x->3

Which looks way better than your LaTeX!! No?

----------------------

Ok, now, back to your problem, if the expression is not in any of the Indeterminate Forms (0/0, inf/inf, inf - inf,...), you can just simply plug the value in.

$$\lim_{x \rightarrow 3} \tan \left( \frac{\pi x}{4} \right) = \tan \left( \frac{3}{4} \pi \right) = -1$$

Some other examples would be:

$$\lim_{x \rightarrow 5} x ^ 2 + x = 5 ^ 2 + 5 = 30$$

$$\lim_{x \rightarrow 5} \sin (x) = \sin (5)$$

$$\lim_{x \rightarrow \pi} \frac{\sin (x)}{x} = \frac{\sin (\pi)}{\pi} = \frac{0}{\pi} = 0$$

Last edited: Jan 31, 2008
8. Jan 31, 2008

### duki

So sorry about the LaTeX... I'll look over the link before posting again.

I really shouldn't have posted that last question; my calculator was in degree mode and was giving me .0411 as an answer. Sorry for your time.

Thanks for all the help!!
I have a few more questions, but I'm going to look over that link first.

9. Jan 31, 2008

### duki

Ok, we're starting to look at infinite limits.
To be honest, I'm completely confused here...

One of the examples is $$y=\frac{x}{x^2-5x+4}$$

So we find the domain to be all reals where x is not 1 or 4... will this be the case for all equations like this? How do you know when to do it this way, and when to do it the other way, where a limit actually exists? How do you know when a limit doesn't exist?

Then the part comes to find the limits, which require four equations. How do you determine the limits from the negative approach and the positive approach to be infinity? How are these equations different from the previous examples?

Thanks for you patience! :)

10. Feb 1, 2008

### VietDao29

The limit does not exists in two cases below:

The first case, is that the left-hand sided limit, and the right-hand sided limit are different. i.e the limit when x approaches some value a from above (x is close to a, and x > a), and from below (x is close to a, and x < a) are not the same.

You know this limit: $$\lim_{x \rightarrow 0} \frac{\sin x}{x} = 1 , \quad x \mbox{ in radians}$$, right? That means, No matter how x tends to 0 (above, or below), sin(x)/x will tend to 1.

The limit when x approaches 0, of the expression $$\frac{|\sin x|}{x}$$ does not exist, because its two one-sided limits are different.

As x approaches 0 from above (when x is close to 0 enough from above, we can restrict x to be on the interval (0, pi/2), so that sin(x) is positive), so we have:

$$\lim_{x \rightarrow 0 ^ +} \frac{|\sin x|}{x} = \lim_{x \rightarrow 0 ^ +} \frac{\sin x}{x} = 1$$

Whereas, as x approaches 0 from below (when x is close to 0 enough from below, we can restrict x to be on the interval (-pi/2, 0), so that sin(x) is negative), that means:

$$\lim_{x \rightarrow 0 ^ -} \frac{|\sin x|}{x} = \lim_{x \rightarrow 0 ^ -} \frac{-\sin x}{x} = -1$$

So, the limit does not exist at x = 0.

Problem:

Ok, see if you can prove that the following limit DNE:

$$\lim_{x \rightarrow 0} \frac{|x|}{x}$$

The second case, is when the limit is infinity. You can either say that the limit is infinity, or the limit DNE (does not exist). Since infinity isn't actually a number.

The infinite limit occurs when the numerator tends to some number not 0, and the denominator tends to 0.

In your example above, the limit does not exist at x = 1, and x = 4. Since, when x tends to 1, or 4, the denominator tends to 0, whereas the numerator tends to 1, and 4 respectively.

You can think like this: When you divide a number (not 0), by some number that is close to 0, its value will get bigger, and bigger.

Say, 1/0.1 = 10, 1/0.01 = 100, 1/0.001 = 1000, ...

Note that, when the numerator, and denominator, both tend to 0, we have the Indeterminate Form 0/0. We must (rationalize, if it has square root, or cube root, ...), then factor, then do the cancellation, and, lastly, evaluate the limit.

Example:

$$\lim_{x \rightarrow 3} \frac{x - 3}{x ^ 2 - 4x + 3}$$

As x tends to 0, both numerator, and denominator tend to 0, so we must factor it.

$$\lim_{x \rightarrow 3} \frac{x - 3}{x ^ 2 - 4x + 3} = \lim_{x \rightarrow 3} \frac{x - 3}{(x - 3) (x - 1)} = \lim_{x \rightarrow 3} \frac{1}{x - 1} = \frac{1}{3 - 1} = \frac{1}{2}$$.

Can you show us the four equations you mentioned about?

You can have a quick look about One-Sided Limit on wiki. It explains things quite well, there are even some graphs for visualization. Just skim through the page, and see if you can get it.

If there's something that troubles you, just post it here. :)

Last edited: Feb 1, 2008