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 limits.

But most of these should be fairly intuitive.

And they are 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 could 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--

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 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-- let me 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 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 approaches

c, minus the limit of g of x as x approaches c.

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

And we also often call it 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 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. This is just

going to be L times M. Same thing,

if instead of having a function here, we had a constant.

So if we just had the limit-- let

me do it in 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 L. So this whole thing

simplifies to k times L.

And we can do the same thing with 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.

And finally-- this is sometimes called the quotient property--

finally we'll look at the exponent property.

So if I have the limit of-- let me

write it this way-- of f of 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

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 raised 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 as L to the r over s power.

So this is equal to 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 property of limits kind of

are the things that you would naturally want to do.

And if you graph some of these functions,

it actually turns out to be quite intuitive.