# Limit properties | Limits and continuity | AP Calculus AB | Khan Academy

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 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.

First!

Second!

Nice !

Are you gonna make the proves?

lim[nth] as n→0

what does "x aproaches c" mean?

very helpful. thank you.

When i first sow this lesom it loocked a bit confusing sinc i calculate using roman numerals so it loocked like the limit as x aproches a 100(C) but you have explaind it mach better then my teacher thenk you

House wouldn't ask that question. He would deduce why Khan is a Youtube Maths teacher. :p

thanks sal

I know division would cause problems.

Thnku Sir

Is there a property for the limit of the logarithm of a function?

Khan Academy is going to save me this semester! Thank you so much!

where is the proof video?