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What I want to do in this video

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is give you a bunch of properties of limits

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and we're not going to prove it rigorously here -

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- in order to have the rigorous proof of these properties

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we need a rigorous definition of what a limit is

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and we're not doing that in this tutorial -

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- we'll do that in the tutorial on the epsilon-delta definition of limit -

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- but most of these should be fairly intuitive

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and they're very helpful for simplifying limit problems

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in the future

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So let's say we know that the limit of some function

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f(x) as x approaches c is equal to L

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and let's say that we also know that the limit of some

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other function, let's say g(x), as x approaches c

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is equal to M

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Now, given that, what would be the limit

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of f(x) plus g(x) as x approaches c?

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Well, and you could look at this visually

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- if you look at the graphs of two arbitrary functions

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you essentially just add those two functions -

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it'll be pretty clear that this is going to be equal to -

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- and once again, I'm not doing a rigorous proof;

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I'm just really giving you the properties here -

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- this is going to be the limit of f(x) as x approaches c

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plus the limit of g(x) as x approaches c

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which is equal to - well, this right over here is -

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(we'll do that in that same color)

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- this right here is just equal to L: it's going to be equal

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to L plus M - this right over here is equal to M

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Not too difficult

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This is often called the Sum Rule

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or the Sum Property of Limits

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and we could come up with a very similar one

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with differences - the limit as x approaches c of f(x) minus g(x)

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is just going to be L minus M

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It's just the limit of f(x) as x approaches c

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minus the limit of g(x) as x approaches c

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So it's just going to be L minus...

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L minus M

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It's often called the Difference Rule

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or the Difference Property of Limits

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and these once again are very, very (hopefully)

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reasonably intuitive

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Now what happens if you take the product of the functions?

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The limit of f(x) times g(x) as x approaches c?

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Well, lucky for us this is going to be equal to

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the limit of f(x) as x approaches c times the limit of g(x) as x approaches c

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Lucky for us, this is kind of a fairly intuitive property of limits

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So in this case this is just going to be equal to -

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- this is L times M

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L times...

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...this is just going to L times M

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Same thing, if instead of having a function here we had a constant

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So if we just had the limit -

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(I'll do that in the same color)

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- the limit of k times f(x) as x approaches c

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where k is just some constant

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This is going to be the same thing as k times the limit

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of f(x) as x approaches c and that is just equal to...

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...this is just equal to L...

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This is equal to L, so this whole thing

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simplifies to k times...

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...k times L

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And we can do the same thing with the differences -

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- this is often called the Constant Multiple Property -

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- we can do the same thing with the differences

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So if we have the limit as x approaches c

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of f(x) divided by g(x), this is the exact

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same thing as the limit of f(x) as x approaches c

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divided by the limit of g(x) as x approaches c

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which is going to be equal to -

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- I think you get it now -

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- this is going to be equal to L over M

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And finally - this is sometimes called the Quotient Property -

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- finally, we'll look at the Exponent Property

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So if I have...

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...if I have the limit of -

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- let me write it this way -

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- of f(x) to some power

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- and actually let me even

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write it as a fractional power -

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to the r over s power,

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where both r and s are integers -

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then the limit of f(x) to the r over s power

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as x approaches c is going to be the exact same

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thing as the limit of f(x) as x approaches c

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raised to the r over s power

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once again, when r and s are both integers

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and s is not equal to zero, otherwise this exponent

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would not make much sense

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and this is the same thing...

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...this is the same thing as L...

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...this is the same thing as L to the r over s power

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This is equal to L to the...

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...L to the r over s power

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So, using these we can actually find the limit

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of many, many, many things and what's neat about it

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is the properties of limits kind of are the things that

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you would naturally want to do and if you

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graph some of these functions actually it

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turns out to be quite intuitive