- [Instructor] Multiple videos and exercises
we cover the various techniques
for finding limits.
But sometimes, it's helpful to think about strategies
for determining which technique to use.
And that's what we're going to cover in this video.
What you see here is a flowchart
developed by the team at Khan Academy,
and I'm essentially going to work through that flowchart.
It looks a little bit complicated at first,
but hopefully it will make sense
as we talk it through.
So the goal is, hey,
we want to find the limit of f of x
as x approaches a.
So what this is telling us to do is,
well the first thing,
just try to substitute what happens when x equals a.
Let's evaluate f of a.
And this flowchart says,
if f of a is equal to a real number,
it's saying we're done.
But then there's this little caveat here.
Probably.
And the reason why is that the limit is a different thing
than the value of the function.
Sometimes they happen to be the same.
In fact, that's the definition of a continuous function
which we talk about in previous videos,
but sometimes, they aren't the same.
This will not necessarily be true
if you're dealing with some function
that has a
point
discontinuity like that
or a jump discontinuity,
or a function that looks like this.
This would not necessarily be the case.
But if at that point
you're trying to find the limit towards,
as you approach this point right over here,
the function is continuous,
it's behaving somewhat normally,
then this is a good thing to keep in mind.
You could just say, hey,
can I just evaluate the function
at that
a over there?
So in general, if you're dealing with
pretty plain vanilla functions like an x squared
or if you're dealing with rational expressions like this
or trigonometric expressions,
and if you're able to just evaluate the function
and it gives you a real number,
you are probably done.
If you're dealing with some type of a function
that has all sorts of special cases
and it's piecewise defined
as we've seen in previous other videos,
I would be a little bit more skeptical.
Or if you know visually around that point,
there's some type of jump
or some type of discontinuity,
you've got to be a little bit more careful.
But in general,
this is a pretty good rule of thumb.
If you're dealing with plain vanilla functions
that are continuous,
if you evaluate at x equals a
and you got a real number,
that's probably going to be the limit.
But I always think about the other scenarios.
What happens if you evaluate it
and you get some number divided by zero?
Well, that case,
you are probably dealing with a vertical asymptote.
And what do we mean by vertical asymptote?
Well, look at this example right over here.
Where we just say the limit
put that in a darker color.
So if we're talking about
the limit
as x approaches one
of one over
x minus one,
if you just try to evaluate this expression
at x equals one,
you would get one over one minus one
which is equal to one over zero.
It says, okay,
I'm throwing it,
I'm falling into this vertical asymptote case.
And at that point,
if you wanted to just understand what was going on there
or even verify that it's a vertical asymptote,
well then you can try out some numbers,
you can try to plot it,
you can say, alright,
I probably have a vertical asymptote here
at x equals one.
So that's my vertical asymptote.
And you can try out some values.
Well, let's see.
If x is greater than one,
the denominator is going to be positive,
and so, my graph
and you would get this from trying out a bunch of values.
Might look something like this
and then for values less than negative one
or less than one I should say,
you're gonna get negative values
and so, your graph might look
like something like that
until you have this vertical asymptote.
That's probably what you have.
Now, there are cases,
very special cases,
where you won't necessarily have the vertical asymptote.
One example of that would be something like
one over x
minus x.
This one here is actually undefined for any x you give it.
So, it would be very,
you will not have a vertical asymptote.
But this is a very special case.
Most times,
you do have a vertical asymptote there.
But let's say we don't fall into either of those situations.
What if when we evaluate the function,
we get zero over zero?
And here is an example of that.
Limit is x approaches negative one
of this rational expression.
Let's try to evaluate it.
You get negative one squared which is one
minus negative one which is plus one
minus two.
So you get zero the numerator.
And the denominator you have negative one squared
which is one
minus two times negative one
so plus two
minus three which is equal to zero.
Now this is known as indeterminate form.
And so on our flowchart,
we then continue to the right side of it
and so here's a bunch of techniques
for trying to tackle something in indeterminate form.
And
likely in a few weeks
you will learn another technique
that involves a little more calculus
called L'hospital's Rule that we don't tackle here
because that involves calculus
while all of these techniques can be done
with things before calculus.
Some algebraic techniques
and some trigonometric techniques.
So the first thing that you might want to
try to do
especially if you're dealing with a rational expression
like this
and you're getting indeterminate form,
is try to factor it.
Try to see
if you can simplify this expression.
And this expression here,
you can factor it.
This is the same thing as
x
x minus two
times x plus one
over
x
this would be x minus three
times x plus one
if what I just did seems completely foreign to you
I encourage you to watch the videos on factoring
factoring polynomials
or factoring quadratics.
And so,
you can see here, alright.
If I make the
I can simplify this 'cause
as long as x does not equal negative one,
these two things are going to cancel out.
So I can say that this is going to be equal to x minus two
over x minus three
for x does not equal negative one.
Sometimes people forget to do this part.
This is if you're really being mathematically precise.
This entire expression is the same as this one.
Because this entire expression is still not defined
if x equals negative one.
Although you can substitute x equals negative one here
and now get a value.
So if you substitute x equals negative one here
even if it's formally taking it away to be
mathematically equivalent,
this would be negative one
minus two
which would be
which would be negative three
over negative one minus three
which should be negative four
which is equal to three fourths.
So if this condition wasn't here,
you can just evaluate it straight up and
this is a pretty plain vanilla function.
Wouldn't expect to see anything crazy happening here.
And if I can just evaluate it at x equals negative one
I feel pretty good.
So once again, we're now going in factoring.
We're able to factor.
We have valued,
we simplify it.
We evaluate the expression
the simplified expression now,
and now we were able to get a value.
We were able to get three fourths,
and so we can feel pretty good that the limit here
in this situation is three fourths.
Now, let's
and I would categorize what we've seen
so far is
the bulk of the limit exercises
that you will likely encounter.
Now the next two,
I would call slightly fancier techniques.
So if you get indeterminate form
especially you'll sometimes see it with radical expressions
like this.
Rational radical expressions.
You might want to multiply by conjugate.
So for example, in this situation right here,
if you just try to evaluate it x equals four,
you get the square root of four minus two
over four minus four
which is zero over zero.
So it's that indeterminate form.
And the technique here, because we're seeing
this radical and a rational expression
let's say, maybe we can somehow get rid of that radical
or simplify it somehow.
So let me rewrite.
Square root of x minus two
over
x
minus four.
Let's say a conjugate,
let's multiply it by the square root of x plus two
over the square root of x plus two.
Once again,
it's the same expression over the same expression.
So I'm not fundamentally changing its value.
And so this is going to be equal to,
well if I have a plus b times a minus b
I'm gonna get a difference of squares.
So it's gonna be square root of x squared
which is,
it's going to be square root of x squared
minus four
over
well square root of x squared
is just going to be
x minus four.
So let me rewrite it that way.
So that's x minus four
over
x minus four
times square root of x plus two,
square root of x plus two.
Well, this was useful because now
I can cancel out x equals four
or x minus four right over here.
And once again, if I wanted it
mathematically to be the exact same expression,
I'd say well, now this is going to be equal to
one
one over the square root of x plus two
for x does not equal four,
but we can definitely see what
this function is approaching
if we just now substitute x equals four
into this simplified expression.
And so, that's just gonna be one over
so if we just substitute
if we just substitute x equals four here
you'd get one over square root of four
plus two
which is equal to one fourth.
And once again, you can feel pretty good
that this is going to be
your limit.
We've gone back into the green zone.
If you're actually to plot
this original function,
you would have a point discontinuity.
You would have a gap at x equals
four
but then when you do that simplification
and factoring out that x minus
or canceling out that x minus four,
that gap would disappear.
So that's essentially what you're doing.
You're trying to find the limit as we approach
that gap which we got right there.
Now, this final one.
This is dealing with trig identities.
And in order to do these,
you have to be pretty adept at your trig identities.
So if we're saying the limit,
I'll do that at a darker color.
So if we're saying the limit
as x approaches zero
of sine of x
over sine
of two x
well, sine of zero zero,
sine of zero zero,
you're gonna be at zero, zero.
Once you get indeterminate form,
we fall into this category,
and now you might recognize this is going to be
equal to the limit
as x approaches zero of sine of x
we can rewrite sine of two x
as two sine
x cosine x
and then those two can cancel out for all x's not
equaling
for all x's not equaling zero
if you want to be really mathematically precise.
And so, there would've been a gap there for sure
on the original graph
if you were to graph y equals this.
But now, for the limit purposes,
you can say this is this limit
is
this limit is going to be the limit
as x approaches zero
of one over two cosine of x.
And now we can go back to
this green condition right over here,
because we can evaluate this at x equals zero.
It's going to be one over two times cosine of zero.
Cosine of zero is one.
So this is going to be equal to one half.
Now in general, none of these techniques work,
and you will encounter few other techniques
further on once you learn more calculus,
then you fall on the base line.
Approximation.
And approximation, you can do it numerically.
Try values really really really close
to the number you're trying to find the limit on.
If you're trying to find the limit as x approaches zero
try 0.00000000001.
Try negative 0.0000001
if you're trying to find the limit is x approaches four
try 4.0000001.
Try 3.9999999999
and see what happens.
But that's kind of the last ditch.
The last ditch effort.