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Studying for a test? Prepare with these 14 lessons on Limits and continuity.

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# Limits of combined functions

Video transskription

- [Voiceover] So let's find the limit of f of x times h of x,
as x approaches zero. Alright, we have graphical
depictions of the graphs, y equals f of x and y equals h of x. And we know from our limit properties, that this is gonna be the
same thing as the limit, as x approaches zero of f of x, times, times the limit as x
approaches zero of h of x. And let's think about
what each of these are, so let's first think about
f of x, right over here. So in f of x, as x approaches zero, notice, the function
itself isn't defined there, but we see when we approach from the left, we are approaching, the
function seems to be approaching the value of negative one, right over here. And as we approach from the right, the function seems to
be approaching the value of negative one. So the limit here, this
limit here, is negative one. As we approach from the left, we're approaching negative one, as we approach from the right, the value of the function seems to be approaching negative one. Now what about h of x? Well, h of x we have down here. As x approaches zero,
as x approaches zero, the function is defined at x equals zero, it looks like it is equal to one, and you could, and the
limit is also equal to one, we could see that as we
approach it from the left, we are approaching one, as we approach from the right, we are approaching one. As we approach x equals
zero from the left, we approach, the function approaches one. As we approach x equals
zero from the right, the function itself is approaching one. And it makes sense that the
function is defined there, and is defined at x equals zero, and the limit as x approaches zero is equal to the same as, is equal to the value of
the function at that point, because this is a continuous function. So this is, this is one, and
so negative one times one is going to be equal to,
is equal to, negative one. So that is equal to negative one. Let's do a few more of these. So we have the limit of
negative two times f of x plus three times h of x, as
x approaches negative three. Well, once again, we can
use our limit properties. We know that this is the
same thing as the limit, as x approaches negative
three of negative two f of x, plus the limit as x
approaches negative three of three times h of x. And this is the same thing as the limit, or I should say this is
equal to negative two times the limit as x
approaches negative three, the limit of f of x, as x
approaches negative three, plus, we can take this scaler out, three times the limit of h of x, as x approaches negative three. And so we just have to
figure out what the limit of f of x is as x
approaches negative three, and the limit of h of x as
x approaches negative three. So I'll first do f of x, so f of x right over here, limit is x approaches negative three, when we approach negative
three from the left-hand side, it seems the value of the
function is approaching three, and as we approach it
from the right-hand side, it seems like the value
of the function is zero. So our left-handed and right-handed limits are approaching different things, so this limit actually does not exist. Does not exist. This limit actually here does exist. But since this limit doesn't exist, and we need to figure out this limit in order to figure out this entire limit, this whole thing does not exist. Does not, does not exist. The limits that it's
made up of need to exist, this is a combined limit, so each of the pieces need to exist in order for the, in this
case, the scaled up sum, to actually exist. Let's do one more. Alright. So these are both, looks
like, continuous functions. So we have the limit is x approaches zero h of x over g of x. So once again, using our limit properties, this is going to be the same thing as the limit of h of x as x approaches zero over the limit of g of
x as x approaches zero. Now what's the limit of h
of x as x approaches zero? This is, let's see, as we
approach zero from the left, as we approach x equals
zero from the left, our function seems to be approaching four, and as we approach x
equals zero from the right, our function seems to be approaching four, and that's also what the
value of the function is at x equals zero, that makes sense, because this is a continuous function, so the limit as we approach x equals zero should be the same as
the value of the function at x equals zero. So this top, this is going to be four, now let's think about the limit of g of x as x approaches zero. So from the left it looks
like, as x approaches zero, the value of the function
is approaching zero. And as x approaches zero from the right, the value of the function
is also approaching zero, which happens to also be,
which also happens to be, g of zero, g of zero is also zero. And that makes sense that the
limit and the actual value of the function at that point is the same, because it's continuous. So this also is zero. But now we're in a strange situation. We have to take four
and divide it by zero. So this limit will not exist, cause we can't take four
and divide it by zero. So even though the limit
of h of x is x equals, as x approaches zero exists, and the limit of g of x as
x approaches zero exists, we can't divide four by zero, so this whole entire limit does not exist. Does, does not exist. And actually, if you were
to plot h of x over g of x, if you were to plot that graph, you would see it even clearer that that limit does not exist, you would actually be able
to see it graphically.