## Limits

Toby Opferman http://www.opferman.net toby@opferman.net Limits Welcome to the tutoiral on limits. This will try to provide a basis for learning what a limit is. A limit is what a function approaches as it approaches some number. Here is an example: limit f(x) x->Infinite That is "The limit of f(x) as x approaches Infinite" If f(x) = 1/x then, 1/2 = .5 1/3 = .33 1/4 = .25 1/100 = .01 .... Notice how the numbers get smaller. Now plug in infinite to x and imagine what number the function f(x) approaches. limit f(x) = 0 x->Infinite The limit is 0. No, the function will never be zero, but it doesn't have to be. It never converges to 1 number, it keeps getting closer and closer to the x axis, closer to 0. Therefore, the limit of the function is 0. Look at this function now: limit x^2 x->Infinite This function has no limit. The limit is infinite. One of the best ways to do a limit is plug in the number. If the number does not exist at that point, then look at numbers before it and after it. If numbers before it go towards a different limit than the ones that go towards it after the number, it is said to have a Left and a Right limit. Let us take the old function again f(x) = 1/x limit 1/x x->0 Plug in 0, and you get undefined. Now, plug in numbers between 1 and 0 1/1 = 1 1/.5 = 2 1/0.000000000001 = 1000000000000 Looks like Infinite from the right, so there is no right limit. Let us try the left. 1/-1 = -1 1/-.5 = -2 1/-0.000000000001 = -1000000000000 Looks like -Infinite from the left, so there is no left limit. Remeber, it must converge to a certain number or BE that certain number for that to be the limit. limit 1/x = 1 x-> 1 The limit to 1/x while x approaches 1 is 1. 1/1 = 1. The limit is either the number it approaches or the number it lands on. Some examples: limit 2/3x = 1/3 x->2 limit |x - 3|/(x - 3) x->3 The limit of the above, you get 0/0 = undefined. If you approach the function from the left: |2 - 3|/(2 - 3) = 1/-1 = -1 |1 - 3|/(2 - 3) = 2/-2 = -1 You see that, From the left it approaches -1. But, if you approach the function from the Right: |4 - 3|/(4 - 3) = 1/1 = 1 |5 - 3|/(5 - 3) = 2/2 = 1 Thus, from the right it approaches 1. We say that the "limit does not exist" But, if we were just looking for a 1 sided limit we could say: Right Limit lim f(x) = 1 x->3 + Left Limit lim f(x) = -1 x->3 - *Notation* The + means approach from the right and the - means approach from the left. It is pretty simple really. It's just finding where a function is continous, and what value does it have at the value it is approaching, or what is the value it is tring to approach.