Limit properties
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- What I want to do in this video
- is give you a bunch of properties of limits
- and we're not going to prove it rigorously here -
- - in order to have the rigorous proof of these properties
- we need a 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 limit -
- - 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(x) as x approaches c is equal to L
- and let's say that we also know that the limit of some
- other function, let's say g(x), as x approaches c
- is equal to M
- Now, given that, what would be the limit
- of f(x) plus g(x) as x approaches c?
- Well, and you could look at this visually
- - if you look at the graphs of two arbitrary functions
- you essentially just add those two functions -
- it'll be pretty clear that this is going to be equal to -
- - and 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(x) as x approaches c
- plus the limit of g(x) as x approaches c
- which is equal to - well, this right over here is -
- (we'll do that in that same color)
- - this right here is just equal to L: it's going to be equal
- to L plus M - this right over here is equal to M
- Not too difficult
- This is often called 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(x) minus g(x)
- is just going to be L minus M
- It's just the limit of f(x) as x approaches c
- minus the limit of g(x) as x approaches c
- So it's just going to be L minus...
- L minus M
- It's often called the Difference Rule
- or the Difference Property of Limits
- and these once again are very, very (hopefully)
- reasonably intuitive
- Now what happens if you take the product of the functions?
- The limit of f(x) times g(x) as x approaches c?
- Well, lucky for us this is going to be equal to
- the limit of f(x) as x approaches c times the limit of g(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 L times M
- Same thing, if instead of having a function here we had a constant
- So if we just had the limit -
- (I'll do that in the same color)
- - the limit of k times f(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(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 differences -
- - this is often called the Constant Multiple Property -
- - we can do the same thing with the differences
- So if we have the limit as x approaches c
- of f(x) divided by g(x), this is the exact
- same thing as the limit of f(x) as x approaches c
- divided by the limit of g(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
- And finally - 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(x) to some power
- - and actually let me 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(x) to the r over s power
- as x approaches c is going to be the exact same
- thing as the limit of f(x) as x approaches c
- raised to the r over s power
- once again, when r and s are both integers
- and s is not equal to zero, 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
- This is equal to L to the...
- ...L to the r over s power
- So, using these we can actually find the limit
- of many, many, many things and what's neat about it
- 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 it
- turns out to be quite intuitive
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
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