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

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.

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