CONCEPTS not MAGIC ( ahh :(

1.- What is the DIFFERENCE between finding the limit of a function at x = c and actually plugging in the number x = c? When are the two cases the SAME?




----OK when finding the limit of a function as x=c, your are looking for an output that gets closer and super close to that number c. And this is shown as lim f(x) as x-->c . On the other hand, when you actually plug in the number c into a determined function, you're looking for exactly a certain output and you are more concerned about what's actually happening at that point c and not at its surroundings.


For example, the function:


f(x)= 2x + 4
by plugging in exactly a number c in this case let's say 3,




f(2)= 2(3) + 4
= 6+ 4
= 10


we get 10 as the exact output at the point 3.


---- They both are the same when the lim of f(x) as x-->c it's equals to f(c). This means that there is not a hole in that part of the function. Therefore, they both are the same when a function is continuous.



2.- What are the SIMILARITIES between finding the derivative and finding the slope of a line? What are the DIFFERENCES between the two?




------The similarities between these two are that for both you are looking for the slope of a line. However, when you are looking for a derivative, you are looking for the slope of a tangent line that touches a certain point at a curve. The slope of that tangent line will give you the slope of the curve at that certain point.

Looking at the graph, the tangent line is the green one that touches that blue curve at a certain point.

And when looking for the slope of a line, you are just looking for the slope of a common line.

LIMITSSS and ideas..

I don't know why but I feel I kind of like limits after all.. because I have to picture some of the problem in my head specially when the limit of x approaches to +- infinity and it makes me understand it better...
HOWEVER....

1.) sometimes when the problems involve sin(x) I get confused. This is because it takes me time to understand how sin(x) as it approaches infinity it's equals 0 .I remember that's the answer but correct me if I'm wrong. I know sin(x) it's always between 1 and -1 but it's not like f(x)= 1/x where the function gets very close to 0 as it goes to infinity...ITS WEIRD.

2.) I also have trouble determining the limits when we have to look at graphs and there is that one filled dot by itself either above or under other open dot that is part of a line. That messes my head up. :-S


3.) Other concept that eludes me it's when we have to find the Right end behavior model and Left end behavior model of a problem like this one y=x^2+e^-x.
First I graph both lines and then I try to figure out to which one each side it's gonna look like but it's really hard. I try not use the calculator but I have toooo...it's sad :(