Algebra of Limits and Limits of Polynomials and Rational Numbers Video Lecture | Mathematics for Grade 11

there are several properties of limits
that enable us to calculate limits
efficiently for example if we have a
constant function for example the
constant function that always outputs a7
then if you take any limit the result
will just be that constant that's what
this rule says another rule is that if
you have the function f of x equals x
that is the function that outputs the
same number of you input then when you
take a limit letting X get closer and
closer to a the outputs also get closer
and closer to a if you take a limit of a
sum you can calculate that from the sum
of the limits we were about to see an
example of this in action similarly if
you have a limit of a product then you
can just evaluate the product of the
limits so here is an example suppose
that we want to calculate the limit as X
approaches 2 for the function x squared
plus 3x because we have a sum inside the
limit one of the properties we just
looked at says that we can calculate
those two limits separately so we'll
have the limit as X approaches 2 of x
squared plus the limit as X approaches 2
of 3x then I can use the other property
that allows us to break up a product so
for example over here where we have the
limit as X approaches 2 of 3 times X I
can change that into the limit as X
approaches 2 of 3 times the limit as X
approaches 2 of X and I can do the same
thing with the x squared here by
thinking of this x squared as x times X
doing so allows me to write this as the
limit as X approaches 2 of x times the
limit as X approaches 2 of X now we can
use those first two properties to finish
simplifying one of the properties said
that since
the function is X that is to say it
outputs the same thing that is input if
X is approaching to the output also
approaches two so both of these limits
approach two so does the very last limit
and then here we have a constant
function as X is approaching to the
outputs approach three they actually
stay at three so if you simplify this 2
times 2 plus 3 times 2 the result is 10
we've just calculated the limit of this
quantity and that's without having to
resort to a graph or a table this
approach is much more efficient than
having to use a graph or a table once we
know how to use these properties let's
go back to our list of properties and
add one more you can also calculate the
limit of a quotient meaning one function
divided by another function and more or
less the same way but there's one point
that we need to be careful about the
limit of the denominator can't be zero
otherwise we'd be dividing by zero and
we know that that's not allowed
so here's another example we can
calculate the limit as X approaches 3 of
X plus 1 over 2x plus 1 by taking the
limit of the numerator divided by the
limit of the denominator and then if you
calculate each of these limits using the
same approach that we did in the last
example you're going to end up with 3
plus 1 for the limit in the numerator
and 2 times 3 plus 1 for the limit in
the denominator and the total result can
be simplified as 4/7
looking at these last two examples we
calculated leads us to a conclusion that
will make it even easier to calculate a
lot of limits the first example was for
a polynomial function that's anything
that can be written in this form the
coefficient times a power of X plus a
coefficient times another power of X and
so on in fact you can think of this last
term as a coefficient a subscript 0
times X to the 0 and it turns out that
any polynomial function is easy to work
with when you're trying to calculate a
limit if you want to calculate the limit
as X approaches a in a polynomial
function all you really have to do is
plug in a rational functions are
functions that are written as one
polynomial divided by another polynomial
the second example we worked was a
rational function and if you go back and
look at that example again you'll see
that the answer we got is what we would
obtain by just plugging in what X is
approaching and we can do that whenever
the denominator is not zero so here's
one more example to illustrate how this
can make things easy for us in this case
we want to calculate the limit as X
approaches zero for 2x squared plus 3x
minus 1 divided by x squared minus 4 and
it turns out all we have to do is plug
in what X is approaching because this is
a rational function the numerator is a
polynomial and the denominator is a
polynomial so if we do that we get two
times zero squared plus three times zero
minus one divided by zero squared minus
four now we do need to make sure that
our denominator is not zero if we're
going to use this approach and in fact
it's not zero squared minus four will
simplify to negative four which is not
zero which means we're fine
if we finish simplify
this becomes negative 1 over negative 4
or 1 4 and that's the final answer
so using the properties of limits can
make it a little bit easier to calculate
a lot of limits first polynomial
functions and rational functions
especially