Limit Examples (part 1) Some limit exercises
Limit Examples (part 1)
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- Welcome back.
- Now that we hopefully have a little bit of an intuition of
- what a limit is, or finding the limit of a function is,
- let's do some problems.
- Some of these you might actually see on your exams or
- when you're actually trying to solve a general limit problem.
- So let's say what is the limit-- once again, my
- pen is not working.
- What is the limit as x approaches-- let's
- say negative 1.
- And let me see, what's a good-- let's say my expression is--
- I'll put it in parentheses so it's cleaner.
- It's 2x plus 2 over x plus 1.
- So the first thing I would always try to do is just say
- what happens if I just stick x straight into this expression?
- What happens?
- Well, what's 2x plus 2 when x is equal to negative 1?
- 2 times negative 1.
- 2 times negative 1 plus 2 over negative 1 plus 1.
- Well, the numerator is negative 2 plus 2-- that equals 0--
- over-- what's the denominator?
- Negative 1 plus 1.
- Over 0.
- And do we know what 0 over 0 is?
- Well, no.
- It's undefined, right?
- So here's a case, just like what we saw in that first
- video, where the limit actually can't equal what the expression
- equals when you substitute x for the number you're trying to
- find the limit of because you get an undefined answer.
- So let's see if, using the limit, we can come up
- with a better answer for what it's approaching.
- Well, since we're just starting with these limit problems,
- let me draw a graph.
- And I think this is going to give you the intuition
- for what we're doing.
- It'll probably give you the answer.
- But then I'll show you how to solve this analytically.
- So if I draw a graph, these are the axes.
- Actually, I'll do the graphical and the analytical
- at the same time.
- So I want to rewrite this expression in a way that
- maybe I can simplify it.
- So 2x plus 2.
- Isn't that the same thing as 2 times x plus 1?
- 2 times x plus 1, right?
- 2x plus 2 is the same thing as 2 times x plus one, and then
- all of that is over x plus 1.
- So as long as this expression and this expression don't equal
- 0, it actually turns out that this function-- let's say
- this is f of x, right?
- This function.
- Well, for every value other than x is equal to negative
- 1, you could actually cancel this and this out.
- And so really, we see that f of x is equal to-- I need to find
- a better tool-- f of x is equal to 2 when x does not
- equal negative 1.
- And we saw when x is equal to negative 1, it's undefined.
- So undefined when equals negative 1.
- So how would we graph that?
- We showed that f of x is equal to 2 when x does not equal
- negative 1 and f of x is undefined when x
- equals negative 1.
- And once again, all I did is kind of rewrite this exact
- same function, right?
- I showed that I could simplify and I could divide the
- numerator and denominator by x plus 1 as long as x does not
- equal negative 1, and that otherwise, it's undefined.
- So let me graph this.
- I'm going to get a different color.
- Maybe I'll go with red.
- So this is 2.
- So we see that x is-- and let me say this is negative 1.
- So for every other value other than negative 1, the value of
- this, of f of x, is equal to 2.
- This is 1, this is 2, this is 3, and so on.
- At negative 1, the graph is undefined.
- So there's a hole there.
- And then we keep going to the left-hand side.
- So if we're going to do the limit, we can just visually
- say, well, as x-- let me do another color now.
- As x comes from the left-hand side, what does f of x equal?
- Well, f of x is 2, 2, 2, 2, 2. f of x is equal to 2
- until we get to exactly negative 1, right?
- And similarly, when we go from the other hand, it's
- the exact same thing.
- f of x is 2, 2, 2 until we get to negative 1.
- So you'll see, and I'll make sure you see it visually here,
- that the limit as approaches negative 1 of 2x plus 2 over
- x plus 1, it equals 2.
- Let me draw a line here so you don't get messed
- up with all of it.
- And I'm not formally, I guess, proving here that the limit is
- 2, but I'm showing you kind of an analytical way, and this is
- actually how it tends to be done in algebra class, is that
- you tend to simplify the expression so that you say, oh,
- if there wasn't a hole here, what would the f of
- x equal, right?
- And then you'd just evaluate it at that point.
- I think this might give you a little intuition, but this
- isn't a formal solution.
- But unless you're asked to, you tend not to be asked
- for a formal solution.
- You actually just tend to ask what the limit is, and this is
- the way you could solve it.
- And actually another way that you could-- I mean, I often
- used to check my answers when I used to do it is you could take
- a calculator and try in-- what happens when: what is f
- of minus 1.001, right?
- And you can also try what is f of negative 0.999, right?
- Because what you want to do is you want to say, well, what
- does the function equal when I get really close to negative 1?
- And then you could keep going closer and closer to negative
- 1 and see what the function approaches, and in this
- case, you'll see that it approaches 2.
- So let's do another problem.
- Well, let's say, what is the limit as x approaches
- 0 of 1 over x?
- I think here it might be useful to draw this graph because
- it'll give you a visual reason, a visual represent-- actually,
- let's do it both ways.
- Let's say-- let's do it the picking-numbers method because
- I think that'll give you an intuition and maybe it'll
- help us draw the graph.
- So let's say that this is f of x.
- f of x-- you can tell my presentation is very
- unplanned-- f of x is equal to 1 over x.
- And we want to find the limit as x approaches 0.
- So what is f of-- actually, let's make a table.
- f of x.
- So clearly when x is equal to 0, we don't know.
- It's undefined.
- 1 over 0 is undefined.
- But what happens when x equals minus 0.01?
- Well, with minus 0.01, 1 over minus 0.01, that is equal
- to negative 100, right?
- What happens when x is equal to minus 0.001, right?
- So we're getting closer and closer to 0 from
- the negative direction.
- Well, here it equals-- make sure my pen is
- working, color right.
- Something's wrong with my tool.
- Now my computer's breaking down.
- Let's see what's going on.
- I think my computer just froze.
- Well, I'm going to try to solve this, and in the very next
- video, I'm going to continue on with this problem.
- So I'll actually see you in the next presentation while I
- figure out why my pen isn't working, and then we'll
- continue with this problem.
- See you very soon.
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