Limit examples (part 3) More limit examples
Limit examples (part 3)
- Let's do some more limit examples.
- So let's get another problem.
- If I had the limit as x approaches 3 of, let's say,
- x squared minus 6x plus 9 over x squared minus 9.
- So the first thing I like to do whenever I see any of these
- limits problems is just substitute the number in and
- see if I get something that makes sense, and
- then we'd be done.
- Well, usually we'd be done.
- I don't want to make these sweeping statements.
- If the function is continuous, we'd be done.
- But if we put the 3 in the numerator, we get 3 squared,
- which is 9, minus 18 plus 9.
- So that equals 0.
- And the denominator also-- let's see, 3 squared minus
- 9, that also equals 0.
- So we don't like having 0/0.
- My pen tool is malfunctioning again.
- So we don't like getting 0, 0, 0, so is there any way we can
- simplify this expression to maybe get it to an expression
- that, when we evaluate it at x equals 3, we actually get
- something that makes sense?
- Well, whenever I see two of these polynomials here, and
- they look, just by inspecting them, relatively easy to
- factor, I like to factor them out because maybe there's the
- same factor in the numerator and the denominator, and
- then we can simplify it.
- So let's say that this is the same thing as-- that looks
- like it's x plus 3-- no, no, no, x minus 3.
- This is x minus 3.
- It actually looks like it's x minus 3 squared, but we're
- just going to write x minus 3 times x minus 3, which is, of
- course, x minus 3 squared.
- And then in the denominator, you know how to factor these,
- this is x plus 3 times x minus 3, all right?
- So the limit as x approaches 3 of this expression is the same
- thing as the limit as x approaches 3 of
- this expression.
- And, of course, there's nothing we can do to change the fact
- that this function, or this expression, is undefined
- at x equals 3.
- But if we can simplify it, we can figure out
- what it approaches.
- Well, if we assume that x is any number but 3, we can cross
- out these two terms because then they wouldn't be 0, right?
- It only is 0 when x is equal to 3 because-- so in the numerator
- and the denominator, we can cross this out.
- And we can say-- and I'm not being very rigorous here, but
- this is kind of how it's taught, and I think you get the
- intuition-- that this is the same thing as the limit as x
- approaches 3 of x minus 3 over x plus 3.
- Now let's just try to stick the x in and see what we get.
- Well, in the numerator, we get 3 minus 3.
- We still get 0.
- But in the denominator here, we get 6, right?
- 3 plus 3 is 6.
- So now we get a good number.
- 0 or 6, well, that's a real number, so it's 0.
- 0/6 is 0.
- So that was interesting.
- The first time we did it, we got the answer 0/0.
- And now we get the answer 0 by simplifying.
- But, of course, it's very important to remember that
- this expression is not defined at x equals 3.
- It's defined everywhere but, but if we were to graph it, and
- I encourage you to do so, you would see that as you get
- closer and closer to x equals 3, the value of this
- expression will equal 0.
- And I know what you're thinking.
- Well, this was 0/0.
- Is every time I get 0/0 going to end up just becoming 0 when
- I evaluate the expression?
- Well, let's explore that.
- Let me clear this.
- Let's say what is-- pen is not working-- the limit as x
- approaches 1 of x squared minus x minus 2.
- No, let's say x squared plus x minus 2.
- As you can see, I do all this in my head, and
- I'm prone to mistakes.
- And all of that over x minus 1.
- Well, once again, if we just evaluate it, let's see what
- happens when x equals 1.
- You get 1 squared plus 1, so it's 2 minus 2.
- You get 0/0.
- So once again, we get 0/0, and we have to do something to
- this maybe to simplify it.
- Well, let's factor the top.
- So that's the same thing as the limit as x approaches 1.
- Well, that's x minus 1 times x plus 2, right?
- And I think you'll often discover when you see a lot of
- limit problems that even if this top factor, if this top
- expression, is hard to factor, chances are, one of the things
- in the denominator that are making this expression
- undefined is probably a factor up here.
- So sometimes you might get a more complex thing that isn't
- as easy to factor as this, but a good starting point is to
- guess that one of the factors is going to be in the bottom
- expression because that's kind of the trick of these problems,
- to just simplify the expression.
- So once again, if we assume that x does not equal 1, and
- this expression would not be 0 and this would not be 0,
- then these two could be canceled out.
- And we get that this is just the same thing as the limit as
- x approaches 1 of x plus 2.
- Well, now this is pretty easy.
- What's the limit as x approaches 1 of x plus 2?
- Well, you just stick 1 in there, and you get 3.
- So it's interesting.
- When we just tried to evaluate the expression at
- x equals 1, we got 0/0.
- And in the previous example, we saw that it evaluated out when
- you simplified it to 0, and in this example, it came out to 3.
- And I really encourage you, if you have a graphing calculator,
- graph these functions that we're doing and see and show
- yourself visually that it's true, that the limit as you
- approach, say, x equals 1 actually does approach the
- limits that were solving for.
- And make up your own problems.
- Hell, that's what I'm doing.
- So you could prove it to yourself.
- So let's do another.
- Let's do one that I think is pretty interesting.
- Let's say what's the limit as x approaches infinity?
- The limit as x approaches infinity of, let's say, x
- squared plus 3 over x to the third.
- So the way I think about these problems as they approach
- infinity, just think about what happens when you get
- really, really, really large values of x.
- And kind of a cheating way of doing this is, if you have a
- calculator, even if you don't have a calculator, put
- in huge numbers here.
- See what happens when x is a million, see what happens when
- x is a billion, see what happens when x is a trillion,
- and I think you'll get the point.
- You'll see what-- if there is a limit here, you'll
- see what it's going to.
- But the way I think about it is, in the numerator, kind of
- the fastest-growing term here is the x squared term, right?
- This is the fastest-growing term here.
- In the denominator, what's the fastest-growing term?
- Well, in the denominator, the fastest-growing term
- is this x to the third.
- Well, what's going to grow faster, x to the
- third or x squared?
- Well, yeah, x to the third's going to grow a lot
- faster than x squared.
- So this denominator, as you get larger and larger and larger
- values of x, is going to grow a lot faster than that numerator.
- So you could imagine if the denominator's growing much,
- much, much faster than the numerator, as you get larger
- and larger numbers, you're going to get a smaller and
- smaller and smaller fraction, right?
- It's going to approach 0.
- And so as you go to infinity, it approaches 0.
- I know that I kind of just hand waved, but that's really
- how you think about it.
- Another way you could do it is you could actually
- divide this fraction.
- You could actually divide this rational expression, and you'll
- get something like 1/x plus something, something,
- something, and then you'd also see, oh, well, the limit as x
- approaches infinity of 1/x is also 0.
- Let's do one more.
- I'll do this fast so I can confuse you.
- The limit as x approaches infinity of 3x squared plus
- x over 4x squared minus 5.
- These problems kind of look confusing sometimes, but
- they're really easy.
- You just have to think about what happens as you get
- really large values of x.
- Well, as you get really large values of x, these small terms,
- these ones that don't grow as fast as these large terms,
- kind of don't matter anymore, right, because you're getting
- really large values of x.
- And this case, these don't matter anymore, and then
- these two x terms grow at the same pace, right?
- And they'll always be kind of growing in
- this ratio of 3 to 4.
- So the limit here is actually that easy.
- It's 3/4.
- So what you do is you just figure out what's the
- fastest-growing term on the top, what's the fastest-growing
- term on the bottom, and then figure out what it approaches.
- If they're the same term, then they kind of cancel out, and
- you say the limit approaches 3/4.
- It's a very nonrigorous way of doing it, but it gets
- you the right answer.
- See you in the next presentation.
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