Limits and Continuity

This is the first big idea in AP Calculus BC, and it lays the basis for most of Calculus. So Calculus is interested in finding the values of things that are infinitesimally small ( like the slope of a point or the area under a curve) and to find these values limits are used. Take for instance the area under the curve, to approximate it you can divide the curve into rectangles, but the area will only be that an approximation. The smaller the width and the more rectangles there were the better the approximation would be. So what calculus does is take the limit of the sum of the rectangles as the width of the rectangles approaches zero ( or the number of rectangles approaches infinity ) which is the exact area. I'll write more on finding areas under curves when I get to big idea 3 Integration.

So a limit of a function as the input approaches some value is really the value the output or the function approaches. So the function doesn't have to be defined at a point to have a limit there.
There are many ways to find limits of function, including using graphs and just plain algebra. Limits can come from two different directions , the negative direction ( or from the left side of the graph , i.e. the value the function approaches from input values less then the value you are taking the limit at ) and the Positive direction ( from the right side of the graph with values larger than the desired input value ). The notation for the two different types of limits looks like this:

lim f(x) = L                                                                            lim f(x) = L     
x→c -                                                                                      x→c +

From the negative direction                                         Positive Direction

So the → means approaches and the x→ c means as x approaches c . The Limits from each side do not have to be the same, and when they are not there is a weird jump in the function.

Some times to approximate limits algebraically you just have to plug in the c value for x in the function and it gives you the limit. But if this limit is undefined ( ∞/∞  or 0/0 ) then you are going to have to try some different methods. If the function has a quotient , then divide the top and bottom by the highest power of x , and then take the limit. There is also a way that you can factor a function out and then cancel expressions leaving a defined limit. A very useful rule for evaluating limits with indeterminate form  ( +∞/∞  or 0/0 ) is L'Hopitals rule which states that if you have a limit in indeterminate form , then you can take the derivative of the top and bottom ( separately  ) and then take the limit of it.

lim    f(x) =+∞/∞  or 0/0                then           lim    f(x) =lim  f '(x)           
x→c g(x)                                                           x→c g(x)  x→c g'(x)     

A function is continuous if ( all of the following conditions are met ):
            lim f(x) = lim f(x) = L                                                        
            x→c -         x→c +

( so the lim f(x)  is defined and equals L )                                                                             
             x→c 

lim f(x) = L =f(c)                                                                               
x→c 

( or in other words the function must be defined at x=c, and the limit of the function as x approaches c must be defined and equal the function's value at x=c )

A point that needs to be covered here is that if the limits of the function from the right hand and the left hand are not the same than the limit of the function at that point is not defined ( the limits from either side have to equal each other for the whole limit to be defined. )

Asymptotes  are places where either the limit as x approaches infinity equals that value ( for y value asymptotes or horizontal asymptotes ) or the limit as x approaches that value equals infinity ( for x values , or vertical asymptotes )

Continuity allows us to have some theorems, like the intermediate value theorem and the extreme value theorem ,and later on the Mean value and Rolles theorem. The Intermediate value theorem states that if a function is continuous on the interval [a,b] then there is some value c that if between the values f(a) and f(b) for the function. The extreme value theorem states that if a function is continuous on an interval it will hit a highest value on an interval. The Means value theorem ( MVT ) and Rolles theorem which is a special case of MVT have to do with derivatives.

There are different types of discontinuity , jump, removable and infinite discontinuities:

jump discontinuities happen when the right hand and the left hand limits don't equal each other ( the function " jumps " at the discontinuity )

removable discontinuous are when the limits equal each other but not the functions defined value.

Infinite discontinuities happen when functions approach positive or negative infinity at a point ( which actually means that their limit and value are undefined )