Limits (Calculus)

I was afraid of this! High school math was a breeze for me, but of course the first problem on the first assignment of my first college math class has me stumped. I feel like I skipped a whole year of math or something.

I checked the solutions manual for the answer, hoping that I could figure out how to work the problem if I knew for sure what the answer was, but it hasn't really helped me any.

This is the graph:

http://img.photobucket.com/albums/v7...e053/graph.jpg

The problems:

A. lim g(x) as x approaches 1
Answer: Does not exist. As x approaches 1 from the right, g(x) approaches 0. As x approaches 1 from the left, g(x) approaches 1. There is no single number L that all the values g(x) get arbitrarily close to as x approaches 1.
(I think I understand this. Maybe)

B. lim g(x) as x approaches 2
Answer: 1

C. lim g(x) as x approaches 3
Answer: 0


Also, when I say "as x approaches 1" I mean the little arrow pointing from the x to the 1 underneath the word "lim." I understood this symbol correctly, right?

My professor didn't explain anything to us. She asked us to read the book, but I can never understand math when reading from the textbook, even when it's stuff I know well.


Anyway, it would help me a lot if someone could help me figure out how to get to these answers x.x

There are two types of limits, from the left and from the right. take your finger and trace the graph starting at x=0 until you get to where x=1. That is the limit from the lft. Then put your finger around where x=2 and go towards where x=1 again, coming from the right to get the limit from the right. If they are different like they are for x=1, then the limit does not exist. If they are the same then the value where your finger is the value of the limit.

For example 3 there is no limit from the right because there is nothing to the right at all, but the limit is where your finger is, not where the actual point is.

That help at all? :p

A little! Thanks n.n

If you have any questions do not be afraid to ask. Limits are very important and the foundation for everything in Calculus.

Also forgot to mention that you are understanding that yes, lim with x arrow number means the limit as x approaches that number. Sometimes there is a plus or minus after the number. A minus means left, a plus means right.

Limits are actually quite simple when it comes down to it. You just have to take the fancy math mumbo-jumbo and dumb it down. Limits ask where the graph appears to be going. So, for 2, you just look at it (tracing with with your finger like qwerty said) and if they both go to the same place. Limit.

The way that helped me understand what was going on though, was the numerical way of looking at a limit. When the problem says x->2, you could, numerically, look at numbers that get closer and closer to two. Here it would look something like this:

x 1.9, 1.99, 1.999, 1.9999
y .9__.99__.999___.9999

x 2.1, 2.01, 2.001, 2.0001
y 1.1, 1.01, 1.001, 1.0001

Looking at the data tables, it's pretty clear that the y value of the graph is getting closer and closer to 1 as you approach two, without knowing what the actual point on the graph is. So, even if the point on the graph was 9, it wouldn't be the limit because all of the numbers around that X Value on the graph clearly get closer to 1, not 9.

Hope my way of learning this helped.

EDIT: The Professor can try and trick you with left and right side limits, these are denounced with a positive or negative mark for an exponent after the x approaches (ex: x->3^-). I'm fairly sure negative is a right-sided limit (limit on the right of the graph approaching the left of the graph) and a positive is a left-sided limit (limit on the left of the graph approaching the right of the graph). Also if the graph looks like it's approaching infinity, like this only both graphs going straight up or down:

http://www.revisioncentre.co.uk/gcse/maths/1overx.gif

There is no limit, if you think about it logically it doesn't make sense for a point on the graph to equal infinity as infinity is not a set number.

Quote:

---

Originally Posted by qwertysaur

Limits are very important and the foundation for everything in Calculus.

---

Protip: you'll stop using them until Calculus II.

EDIT: Unless your professor actually cares about a bunch of formal definitions about theorems. The only ones you really need are the ones that can guarantee a zero.

Thanks, everyone n.n

So I thought I understood enough to at least finish the assignment, but then I see x₀ and I can't figure out what that is >:I

x₀ is just a name for an x value :p

Quote:

---

Originally Posted by qwertysaur

x₀ is just a name for an x value :p

---

I never understood why books like to teach you something, and then replace whatever they explain the problems with crazy variables, if you call it h at first, don't change h to delta x on the first question, how is that teaching? >:0

Quote:

---

Originally Posted by Goldenboko

Quote:

---

Originally Posted by qwertysaur

x₀ is just a name for an x value :p

---

I never understood why books like to teach you something, and then replace whatever they explain the problems with crazy variables, if you call it h at first, don't change h to delta x on the first question, how is that teaching? >:0

---

Because you need to able to change your terminologies quickly; it is an important skill in the technical fields. Math, physics, chemistry, and just about everything else have different variables for the same term. For example, in physics, "F" means "force," and that could be either from an applied load or an electric field. However, in engineering, "P" is a force, and it is only an applied/normal load; it's more likely to be a value hidden in another value, e.g., stresses (same units as pressure, force/area), which can be all sorts of directions. An even more frightening example is simple harmonic motion (pendulums). Generally, you can represent its movement with A*cos(w*t+phi). In math, it's often represented as a second order differential equation, e.g., t''+mt'+bt=something.

It's a skill set that you have to develop. Sure, you can come from one school of thought, that says that we should use Newton and only Newton's calculus notation. Others are Leibniz loyalists. You need to be able to switch from one to the other, especially since either notation has its conceptual/symbolic advantages. Hell, in a less formal environment, maybe your friend is writing a program that can solve all of your equations for you (lol MATLAB "solve"), and you need to be able to quickly decipher all of that junk.

Also, your professors hate you and they're trying to weed out the dumbasses from the slightly less dumbasses. As an engineer, you should not let your ego develop any further: the point isn't that you get things right all the time, it's so that you get things wrong less often than everyone else. You don't have to be right, just make sure that you aren't wrong.

I still think they could change the variables after giving you a chance to understand what they're supposed to represent. It took four semesters of math classes for that to finally sink in.

Thanks for the help, everyone! I would have been totally lost in the lecture today but I kept up. Until I got extremely tired and started falling asleep anyway.

No problem. If you run into any more trouble just post here. :)