Professor Dave here, let’s talk about limits.
The development of calculus arose from discussing limits.
We defined some activity and allowed a parameter to either increase to infinity, like the number
of sides in a polygon, or decrease to zero, like the amount of time elapsed between two
positions, so that we could see what happens as we approach that value.
But the use of limits has been largely replaced by a more modern framework, and we will be
operating out of that framework as we learn about derivatives and integrals.
However, there are select moments where we will be using limits, and we need to understand
what they mean and how to manipulate them, so before moving forward, let’s talk a little
bit about limits, and the laws associated with them.
Quite simply, we can look at some function, and describe a limit of that function as x
approaches some value.
If we have f of x equals x squared, and we want to find the limit as x approaches two,
that will simply be four.
Whether we approach from the negative direction or the positive direction, as we get closer
and closer to an x value of two, we get closer and closer to a value of four for the function.
So the limit of f of x as x approaches two equals four.
One might wonder why we don’t simply evaluate the function at x equals two, and get four.
This is a totally valid question, and in this example, there is no difference.
That’s because this is a continuous function.
The limit of the function as x approaches A is f of A. But some functions have vertical
asymptotes, and we will want to know what a function does as it approaches that asymptote
from either side, but we can’t just plug in the value where the asymptote occurs, as
for that input, the function will be undefined.
In other cases, perhaps there is no asymptote, but there is a point discontinuity.
Let’s remove the point (two, four) from our previous example.
Now, what we described about the limit of the function as x approaches two is no different
than what we had before, even though the function no longer equals four at x equals two.
This is because the function still does the same thing as we approach two from either direction.
Similarly, we may want to know about the end behavior of a function, as x proceeds infinitely
in either direction, and it may not always be obvious from looking at the function in
equation form.
So this is the realm where limits have their application.
Say we are now looking at the function (x minus one) over (x squared minus one).
As we may recall from algebra, this denominator can be factored, and then this (x minus one)
term can cancel, so this function will look exactly like one over (x plus one), which
we can graph quite easily.
But there is one difference.
The original function must have a discontinuity at x equals one, because the denominator is
zero when x equals one, which makes the function undefined.
This is an example where limits become useful, because we want to know what the function
does all the way up until we get to one, from either side, and we can’t just plug the
value directly into the function like we could with the x squared function.
Instead, we can plug in values that get closer and closer to one, both larger and smaller,
and we can clearly see that the function approaches 0.5, or one half.
So that’s the limit of this function as x approaches one.
Even if we change this function so that at precisely x equals one we give the function
some other value, like two, that won’t change the limit, because the limit doesn’t look
at what the function is doing at that point, only what it is doing while approaching that
point, and the behavior of the function everywhere else has remained the same.
Let’s look at something that would be a little tougher to graph.
Take something like sine of x over x.
What is the value of this function as x approaches zero, from either direction?
Well let’s plug in some numbers.
Now we can’t just plug in zero, because the sine of zero is zero, and the denominator
would also be zero.
Having zero over zero is a big problem, because we have no idea what the function is doing
at this point from this information alone, it could be doing anything.
Instead, let’s plug in values approaching zero from either direction, and we will see
that the function is approaching one, so the limit of this function as x approaches zero, is one.
This is an impressive achievement, because using limits, we are able to see that something
that initially looked like zero over zero, was actually equal to one, so we ought to
see the utility of this method.
Let’s also point out that the limit of a function may depend on the direction one is moving.
Take this function, which is defined to be equal to zero if x is less than zero, and
equal to one if x is greater than or equal to zero.
What is the limit of this function as x approaches zero?
Well we get a different value if we are coming at zero from the positive side or negative
side, so there is no limit at this value.
But we can instead describe two different one-sided limits.
The limit of f of x as x approaches zero from the negative side, which is indicated by a
negative sign in superscript over the zero, is zero.
We are getting closer and closer to zero, or rather in this case, the function remains at zero.
The limit of f of x as x approaches zero from the positive side, which is indicated by a
positive sign in superscript over the zero, is one.
We are getting closer and closer to one, or again, in this case, the function remains at one.
These are called left-hand and right-hand limits.
Now what about manipulating limits algebraically?
This can be done, and there are some important laws that we can learn.
First, the limit of a sum of functions is equal to the sum of the limits.
This should sound reasonable, as if one function approaches six and another approaches five,
then the sum of these functions should approach eleven, which is the sum of the individual limits.
As obvious as this sounds, it’s important to make careful sense of these things.
So the limit of a sum is equal to the sum of limits.
The same goes for the limit of a difference, we just switch the sign.
We can also do the same thing for the limit of a product of functions, as well as a quotient,
provided that the bottom limit does not equal zero.
And lastly, the limit of a constant times a function equals the constant times the limit.
So we could look at two functions on the coordinate plane, and describe particular limits at key
points, such as points with discontinuities.
We already know how to find these limits for an individual function, so this application
is no different, we are just doing algebra with limits.
If we can find the limit of each separate function as it approaches two, then we can
find the limit of f of x plus g of x as x approaches two.
Or f of x minus three g of x.
Or f of x times g of x.
Or f of x over g of x.
To these laws we can add just a couple more, all of which require that n be a positive integer.
The limit of a function raised to some power, n, is equal to the limit raised to that power.
Similarly, the limit of some nth root of a function is equal to that root of the limit.
The limit of x to the n as x approaches A, is A to the n, and the limit of the nth root
of x as x approaches A, is the nth root of A. And the last two laws we will mention,
the limit of a constant is equal to that constant, regardless of the parameters of the limit,
and the limit of x as x approaches a particular value is that value.
That seems like a lot of laws, but they come in very handy when trying to assess the limit
of a more complex function.
Take something like (x cubed plus two x squared minus one) over (five minus three x).
Say we want to find the limit of this function as x approaches negative two.
Now that we have these laws, we don’t have to graph the function and inspect it visually,
as that would take a while.
Instead let’s apply the laws.
We have a quotient, so we can just evaluate the limit of the numerator and the limit of
the denominator separately, because of the law we know regarding quotients.
Furthermore, we can split each of those up into individual terms, because of the laws
regarding sums and differences.
And lastly, any constants operating on variables can be pulled outside of the limit, because
of the law we know regarding constants.
We may be left with lots of limits, but all of them are easy to evaluate.
For x cubed, we get negative eight.
For x squared, we get four, times two is eight.
And one is a constant, so that remains one.
On the bottom, five remains five, and x is negative two, times three is negative six,
so after doing some arithmetic, we end up with negative one eleventh.
Now it is true that in this case, we could have just evaluated the function at x equals
negative two to get this answer, but this won’t always be the case, so we should get
used to using limit laws in this manner.
Furthermore, we may also have to do some algebra to an expression first, in order to be able
to use the limit laws, which may involve cancelling terms, or some other manipulation.
So with that, we have a reasonable idea of what limits are and how to calculate them.
We will continue to see limits from time to time, but we will now switch gears a bit in
order to understand differentiation outside of the context of limits, so let’s move
forward and learn about derivatives.