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# Limit properties

AP.CALC:
LIM‑1 (EU)
,
LIM‑1.D (LO)
,
LIM‑1.D.1 (EK)
,
LIM‑1.D.2 (EK)

## Video transcript

what I want to do in this video is give you a bunch of properties of limits so we're not going to prove it rigorously here in order to have the rigorous proof of these properties we need the rigorous definition of what a limit is and we're not doing that in this tutorial we'll do that in the tutorial on the epsilon-delta definition of limits but most of these should be fairly intuitive and they're very helpful for simplifying limit problems in the future so let's say we know that the limit of some function f of X as X approaches C is equal to capital L and let's say that we also know that the limit of some other function let's say G of X as X approaches C is equal to capital M now given that what would be the limit of f of X plus G of X as X approaches C well and you can look at this visually if you look at the graphs of two arbitrary functions you would essentially just add those two functions it'll be pretty clear that this is going to be equal to 1 once again I'm not doing a rigorous proof I'm just really giving you the properties here this is going to be the limit of f of X as X approaches C plus the limit of G of X as X approaches C which is equal to well this right over here is we do that in that same color this right here is just equal to L it's going to be equal to L plus L plus M this right over here is equal to M not too difficult this is often called the sum the sum rule or the sum property of limits and we could come up with a very similar one with differences the limit as X approaches C of f of X minus G of X is just going to be L minus M it's just the limit of f of X as X goes to C minus the limit of G of X as X approaches C so it's just going to be L minus L minus M V that's often called the difference rule or the difference property of limits and these once again are very very hopefully something reasonably intuitive now what happens if you take the product of the functions the limit of f of X times G of X as X approaches C well lucky for us this is going to be equal to the limit of f of X as X approaches C times the limit of G of X as X approaches C lucky for us this is kind of a fairly intuitive property of limits so in this case this is just going to be equal to this is L times M L times this is just going to be L times M same thing if we instead of having a function here we had a constant so if we just had the limit let me do that same color the limit of K times f of X as X approaches C where K is just some constant this is going to be the same thing as K times the limit of f of X as X approaches C and that is just equal to this is just equal to L this is equal to L so this whole thing simplifies to K times K times L and we can do the same thing with the difference this is often called the constant multiple property we can do the same thing with differences so if we have the limit as X approaches C of f of X divided by G of X this is the exact same thing as the limit of f of X as X approaches C divided by the limit of G of X as X approaches C which is going to be equal to I think you get it now this is going to be equal to L over m over over m and finally out this is sometimes called the quotient property finally we'll look at the exponent property so if I have if I have the limit of let me write it this way of f of X to some power and actually let me write even write it as a fractional power to the R over s power where both R and s are integers then the limit of f of X to the R over s power as X approaches C is going to be the exact same thing as the limit of f of X as X approaches C raise to the R over s power once again when R and s are both integers and s is not equal to 0 otherwise this exponent would not make much sense and this is the same thing this is the same thing as L this is the same thing as L to the R over s power so this is equal to L to the L to the R over s power so using these we can actually find a limit of many many many things and it's what's neat about is the properties of limits kind of are the things that you would naturally want to do and if you graph some of these functions actually turns out to be quite intuitive
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