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• - [Instructor] Multiple videos and exercises

• we cover the various techniques

• for finding limits.

• 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