Name: Analyzing unbounded limits: mixed function | Limits and continuity | AP Calculus AB | Khan Academy
Uploaded: 2022-07-05T13:51:31.000Z
Duration: 6 min 16 s
Description: Thousands of YouTube videos with English-Chinese subtitles! Now you can learn to understand native speakers, expand your vocabulary, and improve your pronunciation...

is equal to x over one minus cosine of x minus two,

To-infinitive

and they ask us to select the correct description

of the one-sided limits of f at x equals two.

which is the same thing as cosine of zero,

and so the function is not defined at x equals two,

and that's why it might be interesting to find the limit

as x approaches two, and especially the one-sided limit.

So there's actually a couple of ways you could do it.

There is one way you could do this without a calculator

and thinking about the properties of the cosine function,

and if that inspires you, pause the video and work it out,

and I will do that at the end of this video.

like we've done in other example problems.

where you have x, and then you have f of x.

And so for approaching two from values greater than two,

I do not know what cosine of 0.1 is without a calculator.

so this is very, very close to one without getting to one,

Cosine is never going to be greater than one.

The cosine function is bounded between negative one

I'll just write the x, then I don't need the parenthesis.

The cosine function just oscillates between these two values

so this, this thing is gonna be approaching one

It definitely cannot be greater than one,

for how you can just explore the structure here,

and then you could say, "All right, 2.01.

without being one but it's going to be less than one.

is going to be between negative one and one,

and it could even be including those things

I guess you could say approach one from below.

And so you can start to make some intuitions here.

If it's approaching from below, this thing over here,

this whole expression is going to be positive,

The denominator is positive so this whole thing

and you would see that if you have a calculator

but needless to say, this is going to be unbounded

in the positive direction so we're going to be going

towards positive infinity so these two choices have that,

and we can make the exact same argument as we approach x

as we approach two from below, I should say.

and once again, I don't have a calculator in front of me.

You could evaluate these things at a calculator,

and become very clear that these are positive,

they become even larger and larger positive values,

and the same thing would happen if you did 1.9,

because here, you'd be 1.9 over one minus cosine.

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Analyzing unbounded limits: mixed function | Limits and continuity | AP Calculus AB | Khan Academy

approach

expression

matter

description