Algebra VS Calculus the BIG fight

1) Finding the limit at x=c of a function is actually finding the value as x approaches the constant (c) which turns out to be the y value. Now f(c) is looking for the output when x is c I think that's what I'm trying to say. The output doesn't have to be the same as the limit of x=c but if the limit at x=c is the same as f(c) then the function is continuous and it is only continuous when it is like this. Right?...

2) Both the slope and derivative use the change in y/ the change in x formula but for the slope their is no lim unlike the derivitive where the lim is when h->0 so the difference in the equation would be lim h->0 f(2+h)-f(2)/h where h is how many point away from the initial point. Also a slope would the slope (obviously) of a specific line where a derivative would be involved with Tangent lines of curves.

This is what I got probably very arguable but eh screw it I tried...

What's my limit?

1) I always try to avoid any problem involving Sin(x)/x I do not know what exactly to do I get confused to where am I supossed to plug in the limit especially when it is zero

2) The other types of problems that I have trouble with are the problems with fractions in fractions. For instance, [(1/x+1)+(1/2x)]/x these types of problems get me stumble because I forget how to simplify the problem. Then the same thing about plugging in the limit especially when it is zero...damn zeroes.

3) Int type problems just stumble me. I know how they look but I can't graph them without using the calculator and I really do not know how to find the limit at all I try but fail HELP!!!