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# Limit at infinity of a difference of functions

Video transcript

Let's think about the limit
of the square root of 100 plus x minus the square root
of x as x approaches infinity. And I encourage you
to pause this video and try to figure
this out on your own. So I'm assuming
you've had a go at it. So first, let's just
try to think about it before we try to manipulate
this algebraically in some way. So what happens as x gets
really, really, really, really large, as x
approaches infinity? Well, even though this 100
is a reasonably large number, as x gets really large, billion,
trillion, trillion trillions, even larger than that, trillion,
trillion, trillion, trillions, you can imagine that the
100 under the radical sign starts to matter a lot less. As x approaches really,
really large numbers, the square root of
100 plus x is going to be approximately the same
thing as the square root of x. So for really, really,
large, large x's, we can reason that the
square root of 100 plus x is going to be approximately
equal to the square root of x. And so in that reality-- and
we are going to really, really, really large x's. In fact, there's
nothing larger, where you can keep increasing x's,
that these two things are going to be roughly equal
to each other. So it's reasonable to believe
that the limit as x approaches infinity here is going to be 0. You're subtracting
this from something that is pretty similar to that. But let's actually do some
algebraic manipulation to feel better
about that, instead of this kind of hand-wavy
argument about the 100 not mattering as much
when x gets really, really, really large. And so let me rewrite
this expression and see if we can manipulate
it in interesting ways. So this is 100 plus x minus x. So one thing that might
jump out at you, whenever you see one radical minus
another radical like this, is well, maybe we can
multiply by its conjugate and somehow get rid
of the radicals, or at least transform the
expression in some way that might be a little
bit more useful when we try to find the limit
as x approaches infinity. So let's just-- and
obviously, we can't just multiply it by
anything arbitrary, in order to not change the
value of this expression. We can only multiply it by 1. So let's multiply
it by a form of one, but a form of one that helps
us, that is essentially made up of its conjugate. So let's multiply this. Let's multiply this times
the square root of 100 plus x plus the square root
of x over the same thing, square root of 100 plus x
plus the square root of x. Now notice, this of course
is exactly equal to 1. And the reason why we like
to multiply by conjugates is that we can take advantage
of differences of squares. So this is going to be equal
to-- in our denominator, we're just going to have
the square root of 100. Let me write it this
way actually, 100 plus x plus the square root of x. And in our numerator, we have
the square root of 100 plus x minus the square root
of x times this thing, times square root of 100 plus
x plus the square root of x. Now right over here,
we're essentially multiplying a plus
b times a minus b. We'll produce a
difference of squares. So this is going to be equal
to-- this top part right over here-- is going
to be equal to this. Let me do this in
a different color. It's going to be equal to this
thing squared minus this thing, minus that thing squared. So what's 100 plus x squared? Well, that's just 100 plus x. And then what's square
root of x squared? Well, that's just going to be x. So minus x-- and
we do see that this is starting to
simplify nicely-- all of that over the
square root of 100 plus x plus the
square root of x. And these x's, x minus
x, will just be nothing. And so we are left with 100
over the square root of 100 plus x plus the
square root of x. So we could rewrite the
original limit as the limit as x approaches infinity. Instead of this, we've just
algebraically manipulated it to be this. So the limit as x
approaches infinity of 100 over the square root of 100 plus
x plus the square root of x. And now it becomes much clearer. We have a fixed numerator. This numerator
just stays at 100. But our denominator
right over here is just going to
keep increasing. It's going to be unbounded. So if you're just
increasing this denominator while you keep the
numerator fixed, you essentially have a fixed
numerator with an ever- increasing, or a super-large
or an infinitely-large denominator. So that is going to
approach 0, which is consistent with our
original intuition.