Estimating one-sided limits from graphs: asymptote
About this video
Using a graph with a vertical asymptote at x = 6, this video finds one-sided limits. From the left, g(x) grows without bound, so the left-hand limit is said not to exist; from the right, g(x) approaches -3.
You'll be able to
- recognise when a function becomes unbounded as x approaches a value.
- explain why an unbounded one-sided limit is said not to exist.
- estimate a one-sided limit from a graph near an asymptote.
Key words
- unbounded
- Growing larger and larger without ever levelling off at any number.
- left-handed limit
- The value a function approaches as x gets closer to a point from smaller x values.
- right-handed limit
- The value a function approaches as x gets closer to a point from larger x values.
Transcript
- [Voiceover] Over here we have the graph of y is equal to g of x. What I wanna do is I wanna figure out the limit of g of x as x approaches positive six from values that are less than positive six or you could say from the left, from the, you could say the negative direction.
So what is this going to be equal to? And if you have a sense of it, pause the video and give a go at it. Well, to think about this, let's just take different x-values that approach six from the left and look at what the values of the function are. So g of two, looks like it's a little bit more than one.
G of three, it's a little bit more than that. G of four, looks like it's a little under two. G of five, it looks like it's around three. G of 5.5, looks like it's around five. G of, let's say 5.75, looks like it's like nine. And so, as x gets closer and closer to six from the left, it looks like the value of our function becomes unbounded, it's just getting infinitely large.
And so in some context, you might see someone write that, maybe this is equal to infinity. But infinity isn't, we're not talking about a specific number. If we're talking technically about limits the way that we've looked at it, what is, you'll sometimes see this in some classes.
But in this context, especially on the exercises on Khan Academy, we'll say that this does not exist. Not exist. This thing right over here is unbounded. Now this is interesting because the left-handed limit here doesn't exist, but the right-handed limit does.
If I were to say the limit of g of x as x approaches six from the right-hand side, well, let's see. We have g of eight is there, go of seven is there, g of 6.5, looks like it's a little less than negative three. G of 6.01, little even closer to negative three.
G of 6.0000001 is very close to negative three. So it looks like this limit right over here, at least looking at it graphically, it looks like when we approach six from the right, looks like the function is approaching negative three. But from the left, it's just unbounded, so we'll say it doesn't exist.
Video © Khan Academy, used under CC BY-NC-SA. Watch on Khan Academy. Puzzicle is not affiliated with Khan Academy.