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## Limit rules for sums, differences, products, quotients, & composites

# 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. 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.