limits

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Ari Stern:
 * The limit at a point can only exist if the limit from the left = the limit from the right
 * The limit still exists if there is a hole as long as the limit from the left = the limit from the right

Genevieve Tabby: For a limit to exist, both sides of the function must end up at the same point. However, just because the function may not be continuous, (due to whether it has a hole at that point) that doesn't mean that there can not be a limit at that point. The only requirement limits have is that both sides of the function arrive at the same location. This is why a limit still exists if there is a hole. Here's a picture of what i mean: so, there is a limit at x= -4.

Max Stern: Adding to what Genevieve said above, here is example where one cannot take a limit. Below is a graph of a step function, and say delta x is 1 :(Ms. Sweeney: I don't think you have to say that ∆x is one -- it is implied in the fact that it is the greatest INTEGER function)

Here the function is discontinuous at every integer and therefore there are no limits at those points.

Paul: [edit]

Technically, you are able to take the limit of the greatest integer function; it just depends on where you are trying to find the limit. [quote=Max Stern]Here the function is discontinuous at every integer and therefore there are no limits at those points.[/quote] This talks about finding the limit at a given integer, but not non-integers. Therefore, the limit of the greatest integer function can be found as long as what x is approaching is not an integer.

[/edit]

Alex Gerson: In addition, another example of a situation where a limit does not exist is at an asymptote where the function does not approach the same direction from either side of the asymptote. In this example, there is an asymptote at x=-1, and the limit from the right is infinity, while the limit from the left is negative infinity. This results in the limit not existing at the point x=-1.

Sharon Oser: A third situation where the limit does not exist is when the function oscillates like f(x)=sin(1/x)



__WAYS TO FIND THE LIMIT__
 * Subsitution
 * Factoring
 * Graph
 * Conjugate Method then substitution

__PROPERTIES OF LIMITS__

IF lim f(x)=L x→c AND lim g(x) = M x→c  THEN lim f(x)+g(x) = L+M x →c lim f(x)-g(x) = L-M x→c

lim f(x)g(x) = LM x→c lim f(x)/g(x) = L/M (As long as g(c) is not zero) x→c lim Kf(x) = KL (where K is a constant) x→c

lim f(g(x)) = f(M) x→c

Hannah Weinstein:

Limits can be expressed mathematically using limit notation. For example:

lim f(x) x->a

This means the limit of f(x) as x approaches a. To evaluate the limit, replace x with the value it is approaching:

lim f(x) = f(a) x->a

Sam Ostrum: In certain situations a limit exists at a vertical asymptote. A limit at a vertical asymptote can exist when the function goes to either infinite or negative infinite on BOTH sides of the vertical asymptote. When the limit, approaching the asymptote from the positive direction, is infinite and when the limit, approaching the asymptote from the negative direction, is infinite, then the limit at the x coordinate of the asymptote is infinite (the same goes with negative infinite):

Since the limit approaches infinite on both sides of the asymptote, then a limit of infinite does exist at the asymptote's x coordinate.

Kevin Wu: To be technical, however, a vertical asymptote is not a real limit because as the domain approaches that asymptote, the range approaches infinity or negative infinity; since infinity is a concept and not a real number, the limit technically is inexistent where a vertical asymptote exists.

Susie:

To sum up the of above writing, when a limit does not exist: Vertical asymptote Jump Discontinuity Oscillating functions

Marnie:

To sum up the above pictures

Vertical Asymptote Oscillating funtions Jump Discontinuity

Although a hole is not continuous the limit can still exist if approached from the left and the right (see definition)

Marlee-If your stuck, remember MEAN GIRLS at the Mathletes competition! "Limits. Why couldnt i remember anything about limits?(that was the week Aaron cut his hair). What was on the board behind Aarons head? If the limit never approaches anything..? The limit does not exist. The limit does not exist!" =Sample Questions=