Hi everyone!
This is the second video in a short new video series I’m doing about the basics of calculus.
Last time, in our first section, we talked about functions, which are a special type
of equation that model a one-to-one relationship between x and y.
Today we’re talking about limits and continuity.
They’ll be the foundation for the next section in this series, which is all about Derivatives.
Lets get started by talking about limits.
Think of a simple function, like x+3.
The concept of a limit is just this: If we plug in numbers that are close to 1, we’ll
get back numbers that are close to 4.
To express this limit, we write l-i-m, for limit, as x approaches 1, of the function
x+3, is equal to 4.
In general, we say that the limit as x approaches a number, a, of the function f(x), is equal
to L, and L is the limit of the function.
Here’s another example.
Think of the function ((x-2)(x+2))/(x-2).
Because you can cancel an (x-2) from the numerator and denominator, this function will simplify
to the graph of the line x+2.
However, because setting x equal to 2 in the original function will cause the denominator
to be equal to zero, this function is undefined at that point.
Therefore, we have to modify our graph of the line x+2 to show that the function is
undefined when x is 2, so we draw an empty circle to indicate that we don’t know the
functions value.
Keep in mind that this makes the function discontinuous at that point, which will talk
more about later on in this video.
If the function is undefined there, then what can we say about the value at that point?
Well, we can’t give the actual value, but we can give the limit.
If we look at the graph, we can see that the value of the function gets close to 4 as you
approach x=2 from either side.
Therefore, the limit as x approaches 2 of this function is equal to 4.
When it comes to solving for limits, notice that in both of the previous examples we could
have simply plugged in the value we were approaching, and found the limit of the function.
That’s called substitution, or just good old plug and chug.
Limit problems don’t usually work out that easily though.
More often than not, if we try substitution we’ll end up with either zero over zero,
or zero in just the denominator of our function, both of which are undefined.
In those cases, we’ll have to use other techniques to solve for the limit.
Try factoring the numerator and denominator and then canceling terms.
If your function contains a fraction and a non-fraction, try finding a common denominator
and then simplifying.
If your function contains a radical, or square root sign, try multiplying by the conjugate.
No matter what you use, your goal will be to simplify the function enough so that you
eventually get back to substitution.
If none of these work, we’ll have to get a little fancier, but we’ll save those techniques
for another video.
Sometimes you’re asked to find the limit not as you approach a real number, but instead,
as x approaches positive or negative infinity.
If you need to calculate an infinite limit, you can use the following three rules as shortcuts.
If the degree of the numerator and denominator are equal to one another, use ratio of coefficients
to find the limit.
If the degree of the numerator is less than the degree of the denominator, then the infinite
limit will be zero, regardless of whether you approach negative or positive infinity.
If the degree of the numerator is greater than the degree of the denominator, then the
infinite limit will be positive infinity if you approach positive infinity, and negative
infinity if you approach negative infinity.
Now that we understand the basics of limits, let’s take a couple minutes to talk about
continuity.
A function is continuous if there are no holes, breaks, jumps, or gaps of any kind in its
graph.
You can also think about it this way: A function is continuous if you can draw the entire thing
without picking up your pen or pencil.
If there is any point on the graph where you have to pick up your pencil off the page and
jump to the next section, the function is discontinuous there.
While we’re at it, let’s take some time to classify the most common types of discontinuity,
most notably point, jump, and infinite discontinuities.
Point discontinuities look like this.
They represent specific points on the graph of a function where the function is not continuous.
Point discontinuities are an example of what we call a removable discontinuity.
They’re removable because we can write a simple equation that will define the function
at the point of discontinuity, and this second equation will plug the hole in the graph.
The fact that we can plug the hole makes the discontinuity removable.
Jump Discontinuities are non-removable discontinuities, because you can’t write a simple equation
that fills in the gap.
Jump discontinuities are breaks in the graph that look like this.
Think about walking along on the graph of the function, and having to jump across the
discontinuity to the next section of the graph.
Infinite discontinuities are discontinuities in the graph at asymptotes; vertical, horizontal
and slant.
Infinite discontinuities are also non-removable, because you can’t write a simple function
that will fill the gap and make the function continuous.
Let’s bring this all together and use what we know so far about limits and continuity
to identify what’s happening at each of these five points on this pretty crazy graph
of a function.
At the first point, we have a point discontinuity at x=-6.
The value of the function there is undefined, but the limit of the function as x approaches
-6 is -2.
Whether you approach -6 from the left side or the right side, the function’s value
is approaching -2.
At the second point, we have an infinite discontinuity at x=-3.
The value of the function there is undefined.
Technically, the limit is also undefined, but sometimes we like to say that the limit
is positive infinity, since the graph approaches positive infinity from both sides of -3.
If the graph of the function approached positive infinity from one side of the asymptote, and
negative infinity from the other, then the limit would be truly undefined.
At the third point, the graph has another point discontinuity at x=-1.
The value of the function there is -3 because of the colored circle being shown there, but
the limit of the function is -2, because as you trace the graph with your finger from
both the left and right side, the value you’re approaching is -2, even though the function’s
value is defined as -3.
At the fourth point, we can see that the graph has a jump discontinuity at x=1.
Jump discontinuities require that the graph has different one-sided limits.
Up to now, we’ve only talked about the general limit, but a function can have different left
and right-hand limits.
The general limit only exists if the left and right hand limits are equal, which will
never be the case at a jump discontinuity.
At x=1, the left-hand limit, or the limit as we approach 1 from the negative side, is
equal to 2.
The right-hand limit as we approach from the positive side is equal to 3.
Because these limits aren’t equal, there is no general limit.
The value of the function is equal to 3, since that is the shaded point.
At the fifth and final point, the graph has another jump discontinuity with a left-hand
limit of 3, a right hand limit of 1, and, since these aren’t equal, no general limit.
The value of the function here is 2, since that is the shaded point.
Next time we’re going to use what we’ve learned about limits and continuity to start
talking about derivatives, and we’ll answer the first fundamental question of calculus:
how to find the rate of a change of a function at a specific point.
I’ll see you then.