Limit examples (part 1) Some limit exercises
Limit examples (part 1)
- 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.
Be specific, and indicate a time in the video:
At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger?
Have something that's not a question about this content?
This discussion area is not meant for answering homework questions.
Share a tip
When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831...
Thank the author
This is great, I finally understand quadratic functions!
Have something that's not a tip or thanks about this content?
This discussion area is not meant for answering homework questions.
At 2:33, Sal said "single bonds" but meant "covalent bonds."
For general discussions about Khan Academy, visit our Reddit discussion page.
Here are posts to avoid making. If you do encounter them, flag them for attention from our Guardians.
- disrespectful or offensive
- an advertisement
- low quality
- not about the video topic
- soliciting votes or seeking badges
- a homework question
- a duplicate answer
- repeatedly making the same post
- a tip or thanks in Questions
- a question in Tips & Thanks
- an answer that should be its own question